DC-OPF and Transmission Expansion
GenX can represent the transmission network in two ways, and can expand it in two ways (note that if the problem has no transmission network, then it is run as a "copper plate"). This page explains both, how they interact, and the modeling traps to avoid when they are combined. Note that the user must set NetworkExpansion = 1 to enable these expansion capabilities. In addition, we refer to any expansion or new build as "network expansion" in the documentation.
1. Transport Model vs. DC-OPF
By default (DC_OPF: 0) GenX uses a transport ("pipe-flow") model: each line carries a flow $\Phi_{l,t}$ limited only by the line's available capacity,
\[-\varphi^{cap}_{l} \leq \Phi_{l,t} \leq \varphi^{cap}_{l} \qquad \forall l \in \mathcal{L},\ \forall t \in \mathcal{T}\]
subject to the zonal power balance. Power flow between zones is only constrained by capacity limits and not by any other dynamics.
Setting DC_OPF: 1 adds the linearized DC power-flow physics. A voltage phase angle $\theta_{z,t}$ is introduced for each zone, flow on a line is pinned to the angle difference across it, and the angle difference is itself bounded:
\[\begin{aligned} & \Phi_{l,t} = \mathcal{B}_{l} \times \Big(\sum_{z \in \mathcal{Z}} \varphi^{map}_{l,z}\,\theta_{z,t}\Big) && \forall l \in \mathcal{L},\ \forall t \in \mathcal{T} \\ & -\Delta\theta^{\max}_{l} \leq \sum_{z \in \mathcal{Z}} \varphi^{map}_{l,z}\,\theta_{z,t} \leq \Delta\theta^{\max}_{l} && \forall l \in \mathcal{L},\ \forall t \in \mathcal{T} \\ & \theta_{1,t} = 0 && \forall t \in \mathcal{T} \end{aligned}\]
where $\varphi^{map}_{l,z} \in \{-1, 0, 1\}$ is the network incidence matrix and zone 1 is the slack (reference) bus. The line susceptance is computed from the input data as $\mathcal{B}_{l} = \mathrm{kV}_l^2 / X_l$, with $X_l$ the line reactance in Ohms.
Because DC-OPF constrains how power flows rather than just how much, a DC-OPF solution is always at least as expensive as the transport solution on the same system. That relationship is exploited by the Benders transport hot-start described in Section 6.
Required Network.csv Columns when DC_OPF: 1
| Column | Description |
|---|---|
Line_Voltage_kV | Line voltage in kV. |
Line_Reactance_Ohms | Line reactance in Ohms. Susceptance is derived as $\mathrm{kV}^2/X$. |
Angle_Limit_Rad | Maximum angle difference $\Delta\theta^{\max}_{l}$ across the line, in radians. |
2. Two Kinds of Expansion
NetworkExpansion: 1 turns on transmission expansion. GenX supports two mutually mechanisms, chosen and mutually exclusive per line via the Discrete_Build column of Network.csv.
Every row of Network.csv is an independent line: all per-line inputs are read positionally, and nothing requires the Start_Zone/End_Zone pair to be unique. You can therefore place several rows on the same corridor — the same pair of zones — to model parallel circuits explicitly. This is useful when the parallel lines are not identical: for example two existing lines between the same buses with different susceptances (Line_Reactance_Ohms), capacities, or loss factors, each of which then contributes its own flow, flow–angle coupling, and angle limit under DC-OPF.
This is a different mechanism from Discrete_Build = 1 corridor (below), which generates several identical candidate lines from a single row. Use manually authored rows when you need parallel lines with different electrical properties; use Discrete_Build when you want several interchangeable fixed-size copies of the same new line.
Continuous Reinforcement (Discrete_Build = 0)
The classic GenX behavior. Each eligible line gets a continuous variable $\bigtriangleup\varphi^{cap}_{l} \in [0, \overline{\bigtriangleup\varphi^{cap}_{l}}]$ (vNEW_TRANS_CAP) that adds MW to the existing capacity of that line:
\[\varphi^{cap}_{l} = \overline{\varphi^{cap}_{l}} + \bigtriangleup\varphi^{cap}_{l}, \qquad \bigtriangleup\varphi^{cap}_{l} \leq \overline{\bigtriangleup\varphi^{cap}_{l}}\]
This represents reinforcing an existing corridor — restringing, reconductoring, upgrading terminal equipment — and is a linear decision. The set of eligible lines is those with Line_Max_Reinforcement_MW >= 0.
Discrete New Lines (Discrete_Build = 1)
Set DiscreteInvestments: 1 in genx_settings.yml and mark a corridor with Discrete_Build = 1. GenX then represents expansion on that corridor as one or more fixed-size new lines, each with a binary build decision $x_l \in \{0,1\}$ (vNEW_TRANS_LINES) and each contributing New_Line_Cap_Size_MW of capacity when built:
\[\varphi^{cap}_{l} = \overline{\varphi^{cap}_{l}} + x_{l} \times \overline{F}_{l}\]
This represents building physical new circuits, where you either build a new line or you do not. A new circuit changes the corridor's electrical properties discretely rather than continuously.
How the corridor is expanded at load time. Internally when GenX calls load_network_data!, a corridor flagged Discrete_Build = 1 is rewritten into several rows before the model is built, such that each discrete new addition is added to the internal data as a new line. The number of new lines is the floor of the maximum reinforcement value divied by the new line capacity size:
n_new = floor(Line_Max_Reinforcement_MW / New_Line_Cap_Size_MW)If there is existing capacity defined in the network CSV, this is treated as its own line which cannot have continuous expansion. If the corridor has no existing capacity, no such row is created at all and the corridor consists solely of its candidate lines (see scenario 5 below for why). Lines marked with Discrete_Build = 1 are not able to expand continuously. A corridor cannot be both continuously reinforced and discretely expanded. If Line_Max_Reinforcement_MW is not an integer multiple of New_Line_Cap_Size_MW, GenX warns and rounds down.
Symmetry breaking for discrete new lines. Parallel candidate lines on the same corridor (i.e., generated from the same line of the Network.csv) are identical, so any subset of size $k$ is an equivalent solution — a degeneracy that could slow down branch-and-bound. GenX imposes a build order on each group of more than one identical candidate line (DISCRETE_BUILD_LINE_GROUPS):
\[x_{g_{i-1}} \geq x_{g_{i}} \qquad \forall i \geq 2\]
A user is free to use multiple rows of the Network.csv file for the same corridor, such as if they want to have an existing line with continuous expansion available and an option to build a parallel discrete line.
Because GenX now supports both continuous and discrete line expansion, users will want to be deliberate in how they define this data, perhaps especially in costs. For instance, continuous reinforcement could be thougth of as reconductoring, which may be cheaper than building a new line. In this way, both reconductoring and new line constructions are both supported in GenX, but the use will need to set data deliberately to handle these cases.
Required Network.csv columns when DiscreteInvestments: 1
| Column | Description |
|---|---|
Discrete_Build | 0/1. 1 = this corridor hosts discrete fixed-size new lines. |
New_Line_Cap_Size_MW | MW capacity of each discrete new line. Required (errors if missing). |
BigM | Big-M for the flow–angle coupling on candidate lines. Only used when DC_OPF: 1 and Bilinear_DC_OPF: 0. Defaults to 10 × New_Line_Cap_Size_MW if absent. |
DiscreteInvestments: 1 also enables integer capacity builds for resources, via a Discrete_Build column in the resource CSVs (e.g. Thermal.csv). GenX warns for each input file that is missing the column when the setting is on.
3. Combining DC-OPF with discrete builds
This is where the two features interact, and it is the reason discrete builds exist.
Under DC-OPF, an unbuilt candidate line must not impose any physics. GenX splits the lines into two disjoint sets and treats each differently:
| Set | What it is | Flow–angle coupling | Angle limit |
|---|---|---|---|
DISCRETE_BUILD_LINES | Candidate lines with binary $x_l$ | Build-gated | See below |
FIXED_LINES | Every other line | Exact, always on | Always on |
All parallel lines on a corridor share the same angle difference $\sum_z \varphi^{map}_{l,z}\theta_{z,t}$, so a candidate parallel to a fixed line is already governed by that line's always-on limit and needs no limit of its own. A candidate on a corridor with no fixed line is the only thing that can impose one. GenX identifies those lines as NEW_CORRIDOR_DISCRETE_LINES (a subset of DISCRETE_BUILD_LINES, computed at load time) and gives a build-gated angle limit, relaxed by a big-M when the line is not built so that it binds only once something is actually built on the corridor:
\[-\Delta\theta^{\max}_{l} - M^{\theta}(1 - x_{l}) \leq \sum_{z \in \mathcal{Z}} \varphi^{map}_{l,z}\,\theta_{z,t} \leq \Delta\theta^{\max}_{l} + M^{\theta}(1 - x_{l})\]
The flow–angle coupling itself has two formulations, selected with Bilinear_DC_OPF:
Bilinear_DC_OPF: 0 (default) — big-M relaxation. The coupling is relaxed by $M_l$ (the BigM column) when the line is unbuilt, and the flow is separately forced to zero:
\[\begin{aligned} & -M_{l}(1 - x_{l}) \leq \Phi_{l,t} - \mathcal{B}_{l}\Big(\sum_{z \in \mathcal{Z}} \varphi^{map}_{l,z}\,\theta_{z,t}\Big) \leq M_{l}(1 - x_{l}) \\ & -x_{l}\overline{F}_{l} \leq \Phi_{l,t} \leq x_{l}\overline{F}_{l} \end{aligned}\]
The model stays a MILP. The big-M constraint can be loose in Benders decomposition, and if the value is too small, it can incorrectly constrain a built line. Too large a big M value can cause numerical challenges in the solver.
Bilinear_DC_OPF: 1 — exact bilinear coupling. The coupling is written as the literal product of the build variable and the angle difference:
\[\Phi_{l,t} = \mathcal{B}_{l} \Big(\sum_{z \in \mathcal{Z}} \varphi^{map}_{l,z}\,\theta_{z,t}\Big) \times x_{l}\]
This is exact and needs no big-M, but the model becomes mixed-integer nonlinear, and requires a solver that handles bilinear terms (e.g. Gurobi). Setting Bilinear_DC_OPF: 1 with DC_OPF: 0 is contradictory; GenX warns and forces DC_OPF: 1.
The bilinear coupling makes the constraints quadratic, so the resulting problem is a quadratically-constrained program (QCP) rather than an LP/MILP. You must use a solver that can handle quadratic constraints, and (for Benders) one that can return dual values for them:
- Monolithic solves need a QCP-capable solver such as Gurobi.
- Benders solves make the operational subproblems QCPs whose duals form the optimality cuts. The solver must therefore produce QCP duals. Use a nonlinear solver like Ipopt, or Gurobi with
QCPDualset to1(Gurobi does not compute QCP duals unless this is enabled). Without QCP duals the Benders cuts cannot be generated.
Be aware that this approach can suffer from numerical issues as Gurobi sometimes struggles to return QCPDuals even if it is enabled in settings. However, initial tests suggest that the QCP may return stronger cuts than are achieved via the big M formulation (see this paper).
4. The five corridor scenarios in Network.csv
Every corridor you can express falls into one of five cases. The cases are determined by three columns — Line_Max_Flow_MW (does a line exist today?), Line_Max_Reinforcement_MW (may it grow, and by how much?), and Discrete_Build (does it grow continuously or in whole circuits?). Assume NetworkExpansion: 1 throughout, and DiscreteInvestments: 1 for the discrete cases.
| # | Scenario | Line_Max_Flow_MW | Line_Max_Reinforcement_MW | Discrete_Build | New_Line_Cap_Size_MW |
|---|---|---|---|---|---|
| 1 | Existing line, no expansion | 100 | -1 | 0 | — |
| 2 | Existing line, continuous expansion | 100 | 300 | 0 | — |
| 3 | Existing line, discrete expansion | 100 | 300 | 1 | 100 |
| 4 | New corridor, continuous expansion ⚠️ | 0 | 300 | 0 | — |
| 5 | New corridor, discrete expansion | 0 | 300 | 1 | 100 |
The key to reading this table is that eligibility for continuous expansion is Line_Max_Reinforcement_MW >= 0. Any negative value (-1 by convention) prevents any expansion.
1. Existing line, no expansion
Line_Max_Flow_MW = 100, Line_Max_Reinforcement_MW = -1.
The line is fixed at 100 MW and no investment variable is created
(Setting Line_Max_Reinforcement_MW = 0 instead gives the same physical result, but still creates a vNEW_TRANS_CAP variable with an upper bound of zero. Use -1 to keep it out of the model.)
2. Existing line, continuous expansion
Line_Max_Flow_MW = 100, Line_Max_Reinforcement_MW = 300, Discrete_Build = 0.
The line gets a continuous variable $\bigtriangleup\varphi^{cap}_{l} \in [0, 300]$, so its capacity is $100 + \bigtriangleup\varphi^{cap}_{l} \in [100, 400]$ MW. If DC_OPF = 1, the angle limit constraint is always active. When DC_OPF = 1, this can be thought of as reconductoring an existing corridor.
3. Existing line, discrete expansion
Line_Max_Flow_MW = 100, Line_Max_Reinforcement_MW = 300, Discrete_Build = 1, New_Line_Cap_Size_MW = 100.
Internally, GenX rewrites this row into four separate lines:
| Internal row | Capacity | Role |
|---|---|---|
| residual existing | 100 MW fixed | FIXED_LINE — real line, angle limit always on |
| candidate 1 | 0 or 100 MW | DISCRETE_BUILD_LINE, binary $x_1$ |
| candidate 2 | 0 or 100 MW | DISCRETE_BUILD_LINE, binary $x_2$ |
| candidate 3 | 0 or 100 MW | DISCRETE_BUILD_LINE, binary $x_3$ |
\[\lfloor 300 / 100 \rfloor = 3\]
candidate lines are created. Total corridor capacity is therefore $100 + 100\sum_i x_i$, i.e. one of {100, 200, 300, 400} MW — the same range as scenario 2, but reachable only in whole 100 MW circuits, where each new line has identical data to the existing line (e.g., susceptance, costs). The residual existing row has its Line_Max_Reinforcement_MW internally set to -1, so it receives no continuous reinforcement: the 300 MW budget is spent entirely on the discrete candidates. Symmetry breaking forces $x_1 \geq x_2 \geq x_3$.
4. New corridor, continuous expansion ⚠️
Line_Max_Flow_MW = 0, Line_Max_Reinforcement_MW = 300, Discrete_Build = 0.
Capacity is $0 + \bigtriangleup\varphi^{cap}_{l} \in [0, 300]$ MW. Under the transport model this is perfectly fine and is the normal way to offer a greenfield corridor.
Under DC-OPF it can overconstrain line angle limits because its coupling and angle limit are on in every hour regardless of what is built. If the optimizer builds nothing, then $|\Phi_{l,t}| \leq \varphi^{cap}_{l} = 0$ forces $\Phi_{l,t} = 0$, and the coupling $\Phi_{l,t} = \mathcal{B}_l \Delta\theta_{l,t}$ then forces $\Delta\theta_{l,t} = 0$ so that the buses are forced to have identical phase angles. It is recommended to use discrete new line corridors in this case instead.
5. New corridor, discrete expansion
Line_Max_Flow_MW = 0, Line_Max_Reinforcement_MW = 300, Discrete_Build = 1, New_Line_Cap_Size_MW = 100.
When DC_OPF = 1, this avoids the challenge of unintentionally enforced angle limits:
| Internal row | Capacity | Role |
|---|---|---|
| candidate 1–3 | 0 or 100 MW each | DISCRETE_BUILD_LINE and NEW_CORRIDOR_DISCRETE_LINE, binary $x_i$ |
Note there is no residual row. Unlike scenario 3, a discrete corridor with no existing capacity produces only its candidate lines. A big-M constraint guards the line angle limit so that the line angle limit between the buses only applies when the line is built.
5. Warnings and recommended usage
This is scenario 4 vs. scenario 5 above. If a corridor has no existing line (Line_Max_Flow_MW = 0) and you expand it with continuous reinforcement under DC_OPF: 1, its flow–angle coupling and its $\pm\Delta\theta^{\max}_{l}$ angle limit are enforced in all hours regardless of whether any capacity is built. So a line that does not exist still constrains the phase angles of the two zones it would have connected.
With Bilinear_DC_OPF: 0, BigM must be large enough that it does not create an unintended constraint if a line is not built since $\mathcal{B}_l \Delta\theta_l$ is constrained by $M_l$ in this case. However, its size can impact numerical performance of both the monolithic and Benders solves. The default (10 × New_Line_Cap_Size_MW) is a heuristic, not a guarantee. A BigM that is too small can silently cut off valid solutions; one that is too large can make the MILP slow.
Binary line builds (and integer resource builds) result in a MILP instead of an LP and can increase solution time. Consider using Benders decomposition if the model struggles with tractability, and/or set a consider what MIPGap your problem requires.
6. Benders decomposition with DC-OPF and expansion
Under Benders, the line-build decisions ($x_l$, $\bigtriangleup\varphi^{cap}_{l}$) live in the planning problem while the DC-OPF angle constraints live in the operational subproblems. The subproblems can be infeasible for many early planning solutions (an unbuilt network cannot serve demand), and the discrete nature of the problem can result in long solves times and weak cuts.
GenX provides four settings in benders_settings.yml to assist with tractability. They are described in detail on the Benders Decomposition page; in summary:
| Setting | Effect |
|---|---|
RunTransportModel | Solve the transport relaxation first and reuse its cuts. |
LPTransportHotstart | LP-relax the planning problem's integer variables for the transport pass. |
LPDCOPFHotstart | LP-relax the planning problem's integer variables for the DC-OPF pass. |
RegularizationPostHotstart | Keep level-set regularization on after the hot-start passes. |
The key idea is that the transport model is a relaxation of DC-OPF on the same system. Cuts generated against transport subproblems are therefore valid underestimators of the DC-OPF recourse cost, so GenX can solve the (much easier) transport problem first, keep the accumulated cuts as a warm start, then switch the subproblems to DC-OPF and continue. Each pass is additionally hot-started by first solving its LP relaxation to convergence before restoring integrality. These ideas are outlined in the manuscript available here.
A complete worked example is example_systems/12_IEEE_9_bus_DC_OPF_expansion.