Model Formulation

E4ST.setup_model — Method

setup_model(config, data) -> model

Sets up a JuMP Model for E4ST using config and data.

Parameters

Name	Symbol	Letter
$N_G$	:num_gen	Number of generators
$N_B$	:num_bus	Number of buses
$N_L$	:num_branch	Number of branches
$G = \{1,2,...,N_G \}$	N/A	Set of generator indices
$B = \{1,2,...,N_B \}$	N/A	Set of bus indices
$L = \{1,2,...,N_L \}$	N/A	Set of branch indices
$g \in G$	:gen_idx	Generator index
$b \in B$	:bus_idx	Bus index
$l \in L$	:branch_idx	Branch index
$y \in Y$	:year_idx	Year index
$y_{S_g} \in Y$	:yearonidx	Starting Year index for generator `g`

Variables

These are the decision variables to be optimized over. Can be accessed by model[symbol]

Name	Symbol	Unit	Description
$\theta_{b,y,h}$	`:θ_bus`	Radians	Hourly voltage angle of each bus. Reference buses fixed to 0.0
$P_{G_{g,y,h}}$	`:pgen_gen`	MW	Hourly avg. power generated by each generator
$P_{C_{g,y}}$	`:pcap_gen`	MW	Annual power generation capacity of each generator
$P_{S_{b,y,h}}$	`:plserv_bus`	MW	Hourly avg. power served to each bus

Expressions

Expressions are calculated as linear combinations of variables. Expressions can be 1D, 2D, or 3D arrays or sparse arrays where 1D is Y, 2D is NxY, and 3D is NxYxH. Can be accessed by model[symbol]

Name	Symbol	Unit	Description
$P_{F_{l,y,h}}$	`:pflow_branch`	MW	Hourly avg. power flowing through each branch
$P_{F_{b,y,h}}$	`:pflow_bus`	MW	Hourly avg. net power flowing out of each bus
$P_{U_{b,y,h}}$	`:plcurt_bus`	MW	Hourly avg. power curtailed at each bus
$P_{G_{b,y,h}}$	`:pgen_bus`	MW	Hourly avg. power generated at each bus

Constraints

Constraints can be 1D, 2D, 3D, or 4D arrays or sparse arrays where 1D is Y, 2D is NxY, 3D is NxYxH, 4D is NxYxIxH

Name	Constraint	Symbol	Unit	Description
$C_{PB_{b,y,h}}$	$P_{G_{b,y,h}} - P_{S_{b,y,h}} \geq P_{F_{b,y,h}}$	`:cons_pbal_geq`	MW	Constrain the power flow at each bus
$C_{PB_{b,y,h}}$	$P_{G_{b,y,h}} - P_{S_{b,y,h}} \leq P_{F_{b,y,h}}$	`:cons_pbal_leq`	MW	Constrain the power flow at each bus
$C_{PG_{g,y,h}}^{\text{min}}$	$P_{G_{g,y,h}} \geq P_{C_{b,y}}^{\text{min}}$	`:cons_pgen_min`	MW	Constrain the generated power to be above minimum capacity factor, if given.
$C_{PG_{g,y,h}}^{\text{max}}$	$P_{G_{g,y,h}} \leq P_{C_{b,y}}^{\text{max}}$	`:cons_pgen_max`	MW	Constrain the generated power to be below min(availability factor, max capacity factor)
$C_{PS_{g,y,h}}^{\text{min}}$	$P_{S_{b,y,h}} \geq 0$	`:cons_plserv_min`	MW	Constrain the served power to be greater than zero
$C_{PS_{g,y,h}}^{\text{max}}$	$P_{S_{b,y,h}} \leq P_{D_{b,y,h}}$	`:cons_plserv_max`	MW	Constrain the served power to be less than or equal to load power.
$C_{PC_{g,y}}^{\text{min}}$	$P_{C_{g,y}} \geq P_{C_{g,y}}^{\text{min}}$	`:cons_pcap_min`	MW	Constrain the power generation capacity to be less than or equal to its minimum.
$C_{PC_{g}}^{\text{max}}$	$P_{C_{g,y}} \leq P_{C_{g,y}}^{\text{max}}\quad \forall y = y_{S_g}$	`:cons_pcap_max`	MW	Constrain the power generation capacity to be less than or equal to its minimum for its starting year.
$C_{PL_{l,y,h}}^{+}$	$P_{F_{l,y,h}} \leq P_{L_{l,y,h}}^{\text{max}}$	`:cons_branch_pflow_pos`	MW	Constrain the branch power flow to be less than or equal to its maximum.
$C_{PL_{l,y,h}}^{-}$	$-P_{F_{l,y,h}} \leq P_{L_{l,y,h}}^{\text{max}}$	`:cons_branch_pflow_neg`	MW	Constrain the negative branch power flow to be less than or equal to its maximum.
$C_{PCGPB_{g,y}}$	$P_{C_{g,y}} = 0 \quad \forall \left\{ y<y_{S_g} \right\}$	`N/A`	MW	Constrain the power generation capacity to be zero before the start year. Implemented by fixing the variable.
$C_{PCGNA_{g,y}}$	$P_{C_{g,y+1}} <= P_{C_{g,y}} \quad \forall \left\{ y >= y_{S_g} \right\}$	`:cons_pcap_gen_noadd`	MW	Constrain the power generation capacity to be non-increasing after the start year. Generation capacity is only added when building new generators in their start year.
$C_{PCGE_{g,y}}$	$P_{C_{g,y}} == P_{C_{0_{g}}} \quad \forall \left\{ first y >= y_{S_g} \right\}$	`:cons_pcap_gen_exog`	MW	Constrain unbuilt exogenous generators (with `build_type ∈ (real, exog)`) to be built to `pcap_max` in the first year after `year_on`.

Objective

The objective is a single expression that can be accessed via model[:obj]. In general, we add things to the objective via:

source