Mathematical Formulation

Objective function

The objective function equation of the planning mode is the sum of all the regional costs in addition to the inter-regional tranmission 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

\begin{eqnarray} min: Eq\_{obj} = \sum_{reg} Reg\_{obj}(reg) + Exchange\_{links}\_{obj} \;\;\; \forall reg \in regions \end{eqnarray}

Regional Objective Function

\begin{eqnarray} Reg\_{obj}(reg) = \sum_{year} (1+Discount\_{rate}(year,reg))^{-year} \times \sum_{tech} \bigg[InvCost(reg,year,tech)+FixCost(reg,tech,year)+ DecomCost(reg,tech,year)+VarCost(reg,tech,year)+FixTax(reg,tech,year)+InvTax(reg,tech,year)- InvSub(reg,tech,year)-FixSub(reg,tech,year)+ CO2Cost(reg,tech,year)-InvSalvage(reg,tech,year)\bigg] \;\;\; \forall reg \in regions , \forall year \in years , \forall tech \in technologies \end{eqnarray}

Trades Objective Function

\begin{eqnarray} Exchange\_{links}\_{obj} = \sum_{year} (1+Discount_{rate}(year))^{-year} \times \sum_{link} \bigg[InvCost\_{link}(year,link)+ FixCost\_{link}(year,link)+DecomCost\_{link}(year,link)+ VarCost\_{link}(year,link)+FixTax\_{link}(year,link)+ InvTax\_{link}(year,link)-InvSub\_{link}(year,link)- FixSub\_{link}(year,link)-InvSalvage\_{link}\bigg] \;\;\; \forall year \in years , \forall link \in links \end{eqnarray}

Operation mode

Total Objective Function

\begin{eqnarray} min: Eq\_{obj} = \sum_{reg} Reg\_{obj}(reg) + Exchange\_{links}\_{obj} \;\;\; \forall reg \in regions \end{eqnarray}

Regional Objective Function

\begin{eqnarray} Reg\_{obj}(reg) = \sum_{tech} \bigg[FixCost(reg,tech)+ VarCost(reg,tech)+FixTax(reg,tech)- FixSub(reg,tech)+CO2Cost(reg,tech)\bigg] \;\;\; \forall reg \in regions , \forall tech \in technologies \end{eqnarray}

Trades Objective Function

\begin{eqnarray} Exchange\_{links}\_{obj} = \sum_{link} \bigg[FixCost\_{link}(link)+VarCost\_{link}(link)+ FixTax\_{link}(link)-FixSub\_{link}(link)\bigg] \;\;\; \forall link \in links \end{eqnarray}

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 the salvage values on some of the investments where the operational lifetime of the technology lasts longer than the end of the modelling time horizon and subsidies that are applied to some technologies based on the national policies. 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 the taxes and subsidies have been correspondingly written in the source code for the transmission links.

Investment Cost

The cost required for the new installed capacity of the technologies.

\begin{eqnarray} \forall reg \in regions , \forall tech \in technologies , \forall year \in years: \end{eqnarray}

\begin{eqnarray} \boldsymbol{Inv\_{present}}(reg,tech,year) = \boldsymbol{NewCapcity}(reg,tech,year) \times INV(reg,tech,year) \end{eqnarray}

\begin{eqnarray} Depreciation(reg,tech) = \frac{r(1+r)^n}{(1+r)^n-1} \;\;\; \text{where:} \; n = Economic\_{lifetime}(reg,tech) \;\; r = Interest\_{rate}(reg,tech) \end{eqnarray}

\begin{eqnarray} \boldsymbol{Annuity}(reg,tech,year_k) = Depreciation(reg,tech) \times \boldsymbol{Inv\_{present}}(reg,tech,year) \end{eqnarray}

\begin{eqnarray} \boldsymbol{InvCost}(reg,tech,y) = \sum_{year_k=year+1}^{year+Economic\_{lifetime}+1} (1+Discount\_{rate})^{year-year_k} \times \boldsymbol{annuity}(reg,tech,year_k) \end{eqnarray}

Investment Salvage Value

The revenues calculated at the end of the time horizon for the unused period of the investments whose technical liftime exceeds the modelling horizon.

Fixed Cost

The fixed annual operation and maintenance cost based on the total installed capacity of each technology.

\begin{eqnarray} \boldsymbol{FixCost}(reg,tech,year) = \boldsymbol{TotalCapacity}(reg,tech,year) \times F\_{OM}(reg,tech,year) \;\;\; \forall reg \in regions , \forall tech \in technologies , \forall year \in years \end{eqnarray}

Taxes & Subsidies

Taxes and incentives calculated based on the total investment and fixed cost of each technology.

\begin{eqnarray} \forall reg \in regions , \forall tech \in technologies , \forall year \in years: \end{eqnarray}

\begin{eqnarray} \boldsymbol{InvTax}(reg,tech,year) = \boldsymbol{NewCapacity}(reg,tech,year) \times Investment\_{tax}(reg,tech,year) \times INV(reg,tech,year) \end{eqnarray}

\begin{eqnarray} \boldsymbol{InvSub}(reg,tech,year) = \boldsymbol{NewCapacity}(reg,tech,year) \times Investment\_{sub}(reg,tech,year) \times INV(reg,tech,year) \end{eqnarray}

\begin{eqnarray} \boldsymbol{FixTax}(reg,tech,year) = \boldsymbol{TotalCapacity}(reg,tech,year) \times Fix\_{tax}(reg,tech,year) \times F\_{OM}(reg,tech,year) \end{eqnarray}

\begin{eqnarray} \boldsymbol{FixSub}(reg,tech,year) = \boldsymbol{TotalCapacity}(reg,tech,year) \times Fix\_{sub}(reg,tech,year) \times F\_{OM}(reg,tech,year) \end{eqnarray}

Decommissioning Cost

Cost of dismantling the new capacities installed in the vintage years of the modelling horizon.

\begin{eqnarray} \boldsymbol{DecomCost}(reg,tech,year) = \boldsymbol{DecomCap}(reg,tech,year) \times Decom\_{cost}(reg,tech,year) \;\;\; \forall reg \in regions , \forall tech \in technologies , \forall year \in years \end{eqnarray}

Variable Cost

Annual variable operation and maintenance costs including the cost of consumed fuels.

\begin{eqnarray} \boldsymbol{VarCost}(reg,tech,year) = \boldsymbol{Production\_{annual}}(reg,tech,year) \times V\_{OM}(reg,tech,year) \;\;\; \forall reg \in regions , \forall tech \in technologies , \forall year \in years \end{eqnarray}

Carbon Tax

The taxed dedicated to the amount of CO2 emitted by each technology.

\begin{eqnarray} \boldsymbol{CO2Cost}(reg,tech,year) = \boldsymbol{Production\_{annual}}(reg,tech,year) \times Specific\_{emission}(reg,tech,year) \times Carbon\_{tax}(reg,tech,year) \;\;\; \forall reg \in regions , \forall tech \in technologies , \forall year \in years \end{eqnarray}

Capacity

Accumulated New Installed Capacity

\begin{eqnarray} \boldsymbol{Accumulated\_{NewCapacity}}(reg,tech,year) = \sum_{vintage\_{year}} \boldsymbol{NewCapacity}(reg,tech,vintage\_{year}) \;\;\; \forall reg \in regions , \forall tech \in technologies , \forall year \in years \;\;\; if \; year - vintage\_{year} \leq Tech\_{lifetime}(reg,tech) \end{eqnarray}

Total Installed Capacity

\begin{eqnarray} \boldsymbol{TotalCapacity}(reg,tech,year) = \boldsymbol{Accumulated\_{NewCapacity}}(reg,tech,year) + Residual\_{capacity}(reg,tech,year) \;\;\; \forall reg \in regions , \forall tech \in technologies , \forall year \in years \end{eqnarray}

Decomissioned Capacity

Calculates the annual decommissioning capacities based on the previously installed new capacities in the vintage years of the horizon.

\begin{eqnarray} \boldsymbol{DecomCapacity}(reg,tech,y) = \sum_{vintage\_{year}} \boldsymbol{NewCapacity}(reg,tech,vintage\_{year}) \;\;\; \forall reg \in regions , \forall tech \in technologies , \forall year \in years \;\;\; if \; year - vintage\_{year} \geq Tech\_{lifetime}(reg,tech) \end{eqnarray}

Emission

Calculates the annual CO2 emission based on the annual production of each technology and the exogenous specific emission given by the user per unit of output activity.

\begin{eqnarray} \boldsymbol{CO2\_{equivalent}}(reg,tech,year) = \boldsymbol{Production\_{annual}}(reg,tech,year) \times Specific\_{emission}(reg,tech,year) \;\;\; \forall reg \in regions , \forall tech \in technologies , \forall year \in years \end{eqnarray}

Constraints

Energy balance

Guarantees the balance between the supply and demand sides of the energy system.

\begin{eqnarray} \forall reg \in regions , \forall carr \in carriers , \forall tech \in technologies , \forall year \in years , \forall ts \in timesteps \end{eqnarray}

\begin{eqnarray} \sum_{tech \notin tech\_{Demand}} \boldsymbol{Production}(reg,carr,tech,year,ts) + \sum_{REG} \boldsymbol{Imports}(reg,carr,REG,year,ts) \geq \sum_{tech \notin tech\_{Demand} \& tech\_{Supply}} \boldsymbol{Use}(reg,carr,tech,year,ts) + \sum_{REG} \boldsymbol{Exports}(reg,carr,REG,year,ts) + \sum_{tech \in tech\_{Demand}} \boldsymbol{Demand}(reg,carr,tech,year,ts) \end{eqnarray}

Note

All the technologies within Hypatia have one input carrier or/and one output carrier except for the conversion-plus technologies whose the production and use of each input and output carrier must be calculated from the following equations based on the given input and output carrier ratios given by the user:

\begin{eqnarray} \boldsymbol{Production}(reg,carr,tech,year,ts) = \boldsymbol{Production\_{total}}(reg,tech,year,ts) \times Carrier\_{ratio}\_{output}(reg,carr,tech,year,ts) \end{eqnarray}

\begin{eqnarray} \boldsymbol{Use}(reg,carr,tech,year,ts) = \boldsymbol{Use\_{total}}(reg,tech,year,ts) \times Carrier\_{ratio}\_{input}(reg,carr,tech,year,ts) \end{eqnarray}

Trade balance

Ensures that the amounts of imports and exports among any pair of regions are completely balanced.

\begin{eqnarray} \boldsymbol{Imports}(reg,carr,REG,year,ts) = \boldsymbol{Exports}(REG,carr,reg,year,ts) \;\;\; \forall reg \& REG \in regions , \forall carr \in carriers , \forall year \in years , \forall ts \in timesteps \end{eqnarray}

Resource & Technology Availability

Ensures that the production of each technology does not exceed its available activity based both the technology capacity factor and resource capacity factor.

\begin{eqnarray} \sum_{carr} \boldsymbol{Production}(reg,carr,tech,year,ts) \leq \boldsymbol{TotalCapacity}(reg,tech,year) \times Resource\_{capacity}\_{factor}(reg,tech,year,ts) \times Annual\_{production}\_{per}\_{unitcapacity}(reg,tech) \times Timeslice\_{fraction}(ts) \;\;\; \forall reg \in regions , \forall carr \in carriers , \forall tech \in technologies, \forall year \in years , \forall ts \in timesteps \end{eqnarray}

\begin{eqnarray} \sum_{carr} \sum_{ts} \boldsymbol{Production}(reg,carr,tech,year,ts) \leq Capacity\_{factor}\_{}tech \times \sum_{ts} \bigg[\boldsymbol{TotalCapacity}(reg,tech,year) \times Resource\_{capacity}\_{factor}(reg,tech,year,ts) \times Annual\_{production}\_{per}\_{unitcapacity}(reg,tech) \times Timeslice\_{fraction}(ts)\bigg] \forall reg \in regions , \;\;\; \forall carr \in carriers , \forall tech \in technologies, \forall year \in years , \forall ts \in timesteps \end{eqnarray}

Capacity

Maximum & Minimum Regional Total Capacity

Maximum and minimum allowed annual total installed capacity for each technology in each region based on the defined scenario.

\begin{eqnarray} \forall reg \in regions , \forall tech \in technologies, \forall year \in years: \end{eqnarray}

\begin{eqnarray} \boldsymbol{TotalCapacity}(reg,tech,year) \leq Max\_{totalcap}(reg,tech,year) \end{eqnarray}

\begin{eqnarray} \boldsymbol{TotalCapacity}(reg,tech,year) \geq Min\_{totalcap}(reg,tech,year) \end{eqnarray}

Maximum & Minimum Regional New Capacity

Maximum and minimum allowed annual new installed capacity for each technology in each region based on the defined scenario.

\begin{eqnarray} \forall reg \in regions , \forall tech \in technologies, \forall year \in years: \end{eqnarray}

\begin{eqnarray} \boldsymbol{NewCapacity}(reg,tech,year) \leq Max\_{newcap}(reg,tech,year) \end{eqnarray}

\begin{eqnarray} \boldsymbol{NewCapacity}(reg,tech,year) \geq Min\_{newcap}(reg,tech,year) \end{eqnarray}

Maximum & Minimum Overall Total Capacity

Maximum and minimum allowed annual aggregated total installed capacity for each technology over all the regions based on the defined scenario.

\begin{eqnarray} \forall reg \in regions , \forall tech \in technologies, \forall year \in years: \end{eqnarray}

\begin{eqnarray} \sum_{reg} \boldsymbol{TotalCapacity}(reg,tech,year) \leq Max\_{totalcap}\_{global}(tech,year) \end{eqnarray}

\begin{eqnarray} \sum_{reg} \boldsymbol{TotalCapacity}(reg,tech,year) \geq Min\_{totalcap}\_{global}(tech,year) \end{eqnarray}

Maximum & Minimum Overall New Capacity

Maximum and minimum allowed annual aggregated new installed capacity for each technology over all the regions based on the defined scenario.

\begin{eqnarray} \forall reg \in regions , \forall tech \in technologies, \forall year \in years: \end{eqnarray}

\begin{eqnarray} \sum_{reg} \boldsymbol{NewCapacity}(reg,tech,year) \leq Max\_{newcap}\_{global}(tech,year) \end{eqnarray}

\begin{eqnarray} \sum_{reg} \boldsymbol{NewCapacity}(reg,tech,year) \geq Min\_{newcap}\_{global}(tech,year) \end{eqnarray}

Activity

Maximum & Minimum Regional Production

Maximum and minimum allowed production of each technology in each region based on the defined scenario.

\begin{eqnarray} \forall reg \in regions , \forall tech \in technologies, \forall year \in years: \end{eqnarray}

\begin{eqnarray} \boldsymbol{Production\_{annual}}(reg,tech,year) \leq Max\_{production}(reg,tech,year) \end{eqnarray}

\begin{eqnarray} \boldsymbol{Production\_{annual}}(reg,tech,year) \geq Min\_{production}(reg,tech,year) \end{eqnarray}

Maximum & Minimum Overall Production

Maximum and minimum aggregated production of each technology over all the regions based on the defined scenario.

\begin{eqnarray} \forall reg \in regions , \forall tech \in technologies, \forall year \in years: \end{eqnarray}

\begin{eqnarray} \sum_{reg} \boldsymbol{Production\_{annual}}(reg,tech,year) \leq Max\_{production}\_{global}(tech,year) \end{eqnarray}

\begin{eqnarray} \sum_{reg} \boldsymbol{Production\_{annual}}(reg,tech,year) \geq Min\_{production}\_{global}(tech,year) \end{eqnarray}

Output to Input Activity Ratio

Ensures the relationship between the production and consumption of each technology based on the given efficiency (output/input activity ratio)

\begin{eqnarray} \boldsymbol{Production}(reg,tech,year,ts) = Output\_{input}\_{act}\_{ratio}(reg,tech,year) \times \boldsymbol{Use}(reg,tech,year,ts) \;\;\; \forall reg \in regions , \forall tech \in technologies, \forall year \in years , \forall ts \in timesteps \end{eqnarray}

CO₂ Equivalent Emissions

Regional Emission cap

Ensures that the annual amount of CO2 emisstion emitted in each region does not exceed the given maximum allowed annual carbon emissions.

\begin{eqnarray} \sum_{tech} \boldsymbol{CO2\_{equivalent}}(reg,tech,year) \leq Emission\_{cap}\_{annual}(reg,year) \;\;\; \forall reg \in regions , \forall tech \in technologies, \forall year \in years \end{eqnarray}

Overall Emission cap

Ensures that the aggregated annual amount of CO2 emitted over all the regions does not exceed the maixmum allowed annual values by the user.

\begin{eqnarray} \sum_{reg} \sum_{tech} \boldsymbol{CO2\_{equivalent}}(reg,tech,year) \leq Global\_{emission}\_{cap}\_{annual}(year) \;\;\; \forall reg \in regions , \forall tech \in technologies, \forall year \in years: \end{eqnarray}