Constraints

In the following, all constraints used in AnyMOD are listed. Information includes the name used throughout the model, its dimensions, and a stylized formulation of the constraint itself (see Göke (2020) for details). Within these constraints, variables are printed in bold. In addition, the parameters, variables, and the model part associated with the constraints are listed.

To increase performance, AnyMOD stores constraints within DataFrames instead of using JuMPs native containers. Each dimension is represented by a column and integers in these columns relate to nodes within the hierarchical trees of sets (see printObject on how to export these in a readable format). An additional cns column stores the corresponding constraint. Note that final constraints will look unintuitive, because they are scaled to increase performance and converted to standard form by JuMP.

New constraints beyond those listed here can freely be added to a model by using standard JuMP commands.

Balances

Energy balance

Constraints supply to at least exceed demand.

The energy balance can alternatively be enforced as an equality constraint (see Carriers for details). Instead of having a fixed name the constraint is always assigned the name of the carrier being balanced.

name	carrier name
dimension	$Ts_{dis}$, $R_{dis}$, $C$
instances	dispatch resolution of carrier
formulation	$ \scriptstyle dem \, + \, \sum\bm{use} \, + \, \sum\bm{stExt_{in}} \, + \, \sum \bm{exc_{exp}}\, + \, \sum\bm{trd_{sell}} \, + \, \bm{lss} \, \leq \, $
formulation	$ \scriptstyle \; \; \; \; \sum\bm{gen} \, + \, \sum\bm{stExt_{out}} \, + \, \sum (1-lossExc) \bm{exc_{imp}} \, + \, \sum\bm{trd_{buy}} \, + \, \bm{crt} $
parameter	demand exchange losses
variables	generation and use charging and discharging exchange buy and sell curtailment and loss-of-load
part	balance

Conversion balance

Controls the ratio between used and generated quantities.

name	convBal
dimension	$Ts_{exp}$, $Ts_{dis}$, $R_{dis}$, $Te$, ($M$)
instances	reference resolution of technology expansion time-steps of technology relevant modes of technology
formulation	$ \scriptstyle eff_{conv} (\sum\bm{use} \, + \, \sum\bm{stInt_{out}}) \, = \, \sum\bm{gen} \, + \, \sum\bm{stInt_{in}} $
parameter	conversion efficiency
variables	generation and use charging and discharging
part	technology

Storage Balance

Ensures storage levels comply with charged and discharged quantities.

name	stBal
dimension	$Ts_{exp}$, $Ts_{dis}$, $R_{dis}$, $Te$, $C$, ($M$)
instances	dispatch resolution of carrier expansion time-steps of technology relevant modes of technology
formulation	$ \scriptstyle \bm{stLvl_{t}} \, = \, \frac{\bm{stLvl_{t-1}}}{1\,-\,stDis} \, + \, stInflow \, + \, \frac{\bm{stExt_{in}} \, + \, \bm{stInt_{in}}}{1/effSt_{in}} \, - \, \frac{\bm{stExt_{out}} \, + \, \bm{stInt_{out}}}{effSt_{out}} $
parameter	storage efficiencies storage self-discharge storage inflow
variables	charging and discharging storage level
part	technology

Dispatch restrictions

Conversion capacity restriction

Ensures quantities converted comply with the operated conversion capacity.

The graph-based approach that allows to vary resolution by energy carrier within the same model complicates the formulation of capacity constraints for conversion technologies. What kind of constraints are necessary is specified in the capaRestr of the technology part object (see Göke (2020) on how this is derived). Capacity constraints are either enforced on used or generated quantities. In the latter case, quantities have to be corrected for the respective efficiency since AnyMOD always denotes capacities after efficiency.

name	inRestr	outRestr
dimension	$Ts_{exp}$, $Ts_{dis}$, $R_{dis}$, $Te$
instances	required capacity restrictions according to `capaRestr` field of part
formulation	$ \scriptstyle \frac{\bm{use} \, + \, \bm{stInt_{out}}}{ava_{conv}} \, \leq \, \bm{oprCapa_{conv}} $	$ \scriptstyle \frac{\bm{gen} \, + \, \bm{stInt_{in}}}{eff_{conv} \, ava_{conv}} \, \leq \, \bm{oprCapa_{conv}} $
parameter	conversion availability
		conversion efficiency
variables	generation and use charging and discharging operated conversion capacity
part	technology

Storage capacity restriction

Ensures quantities stored comply with the operated storage-input, storage-output, storage-size capacity.

name	stInRestr	stOutRestr	stSizeRestr
dimension	$Ts_{exp}$, $Ts_{dis}$, $R_{dis}$, $Te$, $C$
instances	dispatch resolution of carrier expansion time-steps of technology
formulation	$ \scriptstyle \frac{\bm{stExt_{in}} \, + \, \bm{stInt_{in}}}{ava_{stIn}} \, \leq \, \bm{oprCapa_{stIn}} $	$ \scriptstyle \frac{\bm{stExt_{out}} \, + \, \bm{stInt_{out}}}{ava_{stOut}} \, \leq \, \bm{oprCapa_{stOut}} $	$ \scriptstyle \frac{\bm{stLvl}}{ava_{stOut}} \, \leq \, \bm{oprCapa_{stSize}} $
parameter	storage availability
variables	operated storage capacity
	charging and discharging		storage level
part	technology

Energy ratio restriction

Ensures used and generated quantities comply with the specified ratios.

name	ratioEnerUse{Fix/Low/Up}	ratioEnerGen{Fix/Low/Up}
dimension	$Ts_{exp}$, $Ts_{dis}$, $R_{dis}$, $Te$, $C$, ($M$)
instances	ratios defined
formulation	$ \scriptstyle \bm{use} \; \begin{smallmatrix} = \\[0.5pt] \geq \\[2pt] \leq \end{smallmatrix} \; ratioEner^{use}_{fix/low/up} \sum \bm{use} $	$ \scriptstyle \bm{gen} \; \begin{smallmatrix} = \\[0.5pt] \geq \\[2pt] \leq \end{smallmatrix} \; ratioEner^{gen}_{fix/low/up} \sum \bm{gen} $
parameter	generation and use ratios
variables	generation and use
part	technology

Exchange capacity restriction

Ensures exchanged quantities comply with the operated exchange capacity.

name	excRestr
dimension	$Ts_{dis}$, $R_{from}$, $R_{to}$, $C$
instances	dispatch resolution of carrier and descendant carriers regions can exchange carrier
formulation	$ \scriptstyle \sum \frac{\bm{exc}}{ava_{exc}} \, \leq \, \bm{oprCapa_{exc}} $
parameter	exchange availablity
variables	exchange operated exchange capacity
part	exchange

Trade capacity restriction

Ensures traded quantities comply with the specified trade capacity.

name	trdBuyCap	trdSellCap
dimension	$Ts_{dis}$, $R_{dis}$, $C$, $id$
instances	trade price defined trade capacity defined
formulation	$ \scriptstyle \bm{trdBuy} \, \leq \, trdBuyCap $	$ \scriptstyle \bm{trdSell} \, \leq \, trdSellCap $
parameter	trade capacity
variables	buy and sell
part	trade

Capacity expansion

Definition of installed capacity

Connects installed capacities to expansion variables and residual capacities.

name	capaConv	capaSt{In/Out/Size}	capaExc
dimension	$Ts_{exp}$, $R_{exp}$, $Te$	$Ts_{exp}$, $R_{exp}$, $C$, $Te$	$Ts_{exp}$, $R_{from}$, $R_{to}$, $C$
instance	superordinate dispatch resolution (usually years)
	spatial expansion resolution of technology		spatial expansion resolution of carrier regions can exchange carrier
	if uses or generates carriers	each stored carrier
formulation	$ \scriptstyle \bm{capa_{t}} \, = \, resiCapa_{t} \, + \, \sum\limits_{t' \in (t \, + \, del, t \, +\,del \, + \, life]} \bm{exp_{t'}} $
parameter	residual capacity technical lifetime construction time
variables	installed capacity capacity expansion
part	technology		exchange

Decommissioning of operated capacity

Ensures operated capacities comply with installed capacities.

Decommissioning behaviour is determined by the decomm argument of the model constructor. For none, the equations listed here are not enforced. Instead operated and installed capacities are identical. When the argument is set to recomm, only the first equation that limits operated capacities to installed capacities is enforced.

Lastly, for decomm both equations apply. The second equation will then ensure, that once decommissioned capacities cannot be re-decommissioned again. The expression $\displaystyle \Delta Resi_{+}$ in the equation denotes any increase of residual capacities from $\displaystyle t-1$ to $\displaystyle t$.

name	oprCapaConv	oprCapaSt{In/Out/Size}	oprCapaExc
dimension	$Ts_{exp}$, $R_{exp}$, $Te$	$Ts_{exp}$, $R_{exp}$, $C$, $Te$	$Ts_{exp}$, $R_{from}$, $R_{to}$, $C$
instance	superordinate dispatch resolution (usually years)
	spatial expansion resolution of technology		spatial expansion resolution of carrier regions can exchange carrier
	if uses or generates carriers	each stored carrier
formulation	$ \scriptstyle \bm{oprCapa} \, \leq \, \bm{capa}$
	$ \scriptstyle \bm{oprCapa_{t}} \, \leq \, \bm{oprCapa_{t\,-\,1}} \, + \, \bm{exp_{t}} \, + \, \Delta Resi_{+} $
parameter	residual capacity
variables	operated capacity installed capacity capacity expansion
part	technology		exchange

Cost equations

Within the cost equations the discount factor is used to discount costs to the present. The discount factor for a year $t$ is computed from the discount rates of the current and the previous years as follows:

$\scriptstyle discFac_{t} = \displaystyle \prod_{t' = t_{0}}^{t}(1 \, + \, rateDisc_{t'})^{-1}$

Expansion cost equation

Determines costs of capacity expansion.

name	costExp{Conv,StIn,StOut,StSize}	costExp{Exc}
dimension	$Ts_{sup}$, $R_{exp}$, $Te$	$Ts_{sup}$, $C$
formulation	$ \scriptstyle \bm{costExp\{...\}} \, = \, \sum discFac \, \cdot \, ann\{...\} \, \cdot \, \bm{exp\{...\}} $
	$ \scriptstyle ann\{...\} \, = \, costExp\{...\} \, \cdot \, \frac{rateExp\{...\} \, (1\,+\,rateExp\{...\})^{lifeEco\{...\}}}{(1\,+\,rateExp\{...\})^{lifeEco\{...\}}\,-\,1}$
parameter	discount rate interest rate expansion cost economic lifetime
variables	expansion expansion cost
part	objective

Operating cost equation

Determines costs of operating capacity.

name	costOpr{Conv,StIn,StOut,StSize}	costOpr{Exc}
dimension	$Ts_{sup}$, $R_{exp}$, $Te$	$Ts_{sup}$, $C$
formulation	$ \scriptstyle \bm{costOpr\{...\}} \, = \, \sum discFac \, \cdot \, costOpr\{...\} \, \cdot \, \bm{oprCapa\{...\}} $
parameter	discount rate operating cost
variables	operated capacity operating cost
part	objective

Variable cost equation

Determines costs associated with quantities dispatched. Costs incurred by emission prices are included in costVarUse.

name	costVar{Use,Gen}	costVar{StIn,StOut}	costVar{Exc}
dimension	$Ts_{sup}$, $R_{exp}$, $Te$	$Ts_{sup}$, $R_{exp}$, $Te$	$Ts_{sup}$, $C$
formulation	$ \scriptstyle \bm{costVar\{...\}} \, = \, \sum discFac \, \cdot \, costVar\{...\} \, \cdot \, \{use,gen,stIn,stOut,exc\} \, \cdot \, 10^3 $
	$ \scriptstyle + \, \sum emFac \, \cdot \, emPrc \, \cdot \, \bm{use} \, \cdot \, 10^3$
parameter	discount rate
	variable cost of technologies		exchange cost
	emission factor emission price
variables	variable cost
	generation and use	charging and discharging	exchange quantities
part	objective

Trade cost equation

Determines costs and revenues from buying or selling carriers on an external market.

name	costTrdBuy	costTrdSell
dimension	$Ts_{sup}$, $R_{exp}$, $C$
formulation	$ \scriptstyle \bm{costTrdBuy} \, = $	$ \scriptstyle \bm{costTrdSell} \, = $
	$ \scriptstyle \sum discFac \, \cdot \, trdBuyPrc \, \cdot \, \bm{trdBuy} \, \cdot \, 10^3 $	$ \scriptstyle \sum discFac \, \cdot \, trdSellPrc \, \cdot \, \bm{trdSell} \, \cdot \, 10^3 $
parameter	discount rate trade price
variables	buy and sell trade cost
part	objective

Curtailment and loss-of-load cost equation

Determines costs of curtailment and unmet demand.

name	costCrt	costLss
dimension	$Ts_{sup}$, $R_{exp}$, $C$
formulation	$ \scriptstyle \bm{costCrt} \, = $	$ \scriptstyle \bm{costLss} \, = $
	$ \scriptstyle \sum discFac \, \cdot \, costCrt \, \cdot \, \bm{crt} \, \cdot \, 10^3 $	$ \scriptstyle \sum discFac \, \cdot \, costLss \, \cdot \, \bm{lss} \, \cdot \, 10^3 $
parameter	discount rate cost of curtailment and loss of load
variables	curtailment and loss-of-load curtailment and loss-of-load cost
part	objective

Limiting constraints

Limiting constraints are largely shaped by the corresponding limit parameters on quantities dispatched, expansion and capacity, and by the emission limit.

The name of the constraint will correspond to the name of the respective parameter. Dimension and formulation depend on the way parameter data was provided, as explained here. Lastly, all constraints are stored in the limit part.