Transmission Expansion Planning
TEP · MIP · Garver's problem
Multi-year · Line Investment · Network Design
Which new transmission lines should be built over the next decade, and which existing corridors should be reinforced? Transmission Expansion Planning (TEP) is the network-design half of long-term power-systems investment — complementary to Generation Expansion Planning. A single transmission line can cost $1–5 million per mile and takes 5 to 10 years to permit and construct. The problem is a mixed-integer program with binary investment decisions and a DC-OPF operational subproblem, first formulated by Garver in 1970 and still an active research area six decades later.
The problem
Long-horizon network investment
When a new wind farm comes online in West Texas, how does the power reach the Houston load center? Existing transmission may be congested; the wind's output may spill in high-wind hours because no line is available to carry it east. The transmission planner's job is to identify the candidate new lines that would relieve the bottleneck, calculate the cost of building each, and pick the portfolio that maximizes social welfare — or equivalently, minimizes the sum of investment plus expected operational cost (congestion rents, curtailment, out-of-merit dispatch).
The mathematical structure is a bilevel investment-operations MIP, with the same two-layer skeleton as GEP. Upper level: binary investment variables $x_\ell \in \{0, 1\}$ for each candidate line $\ell$. Lower level: a DC-OPF dispatch in every period (or representative scenario) respecting Kirchhoff's laws and the line-thermal-limit constraint that is now conditional on whether the line exists: $-F_\ell^{\max} x_\ell \le f_\ell \le F_\ell^{\max} x_\ell$. TEP inherits the polynomial-time tractability of DC-OPF at the operational level but the combinatorial explosion from $2^{|\mathcal{L}_{\mathrm{cand}}|}$ investment portfolios.
Modern TEP is increasingly coupled with generation expansion (joint GEP-TEP), storage deployment, and uncertainty (stochastic, robust, adjustable-robust). The key modern question is how much transmission is needed to integrate the clean-energy transition: NREL's Interconnections Seam study (2020) found that a US-scale seam with 30 GW of inter-regional HVDC transmission could save $2–5 billion per year by 2050, assuming net-zero targets are met. The underlying model is a joint GEP-TEP with hourly dispatch.
Mathematical formulation
MIP with DC-OPF subproblem · disjunctive “big-M”
Notation
| Symbol | Meaning | Units |
|---|---|---|
| $\mathcal{N}$ | Buses | — |
| $\mathcal{L}_{\mathrm{ex}}$ | Existing lines | — |
| $\mathcal{L}_{\mathrm{cand}}$ | Candidate (buildable) lines | — |
| $I_\ell$ | Investment cost of candidate line $\ell$ | $ |
| $F_\ell^{\max}$ | Thermal limit of line $\ell$ | MW |
| $b_\ell$ | Susceptance of line $\ell$ | pu |
| $c_g$ | Generation marginal cost at bus $g$ | $/MWh |
| $D_i$ | Load at bus $i$ | MW |
| $M$ | Big-M constant (disjunctive) | pu |
| $x_\ell$ | Build decision for $\ell \in \mathcal{L}_{\mathrm{cand}}$ | {0,1} |
| $p_g$ | Generation output | MW |
| $\theta_i$ | Bus angle | rad |
| $f_\ell$ | Line flow | MW |
| $u_i$ | Load shedding at bus $i$ (penalty) | MW |
Objective
Minimize investment cost plus expected operational cost plus load-shed penalty:
where $s$ indexes operational scenarios (typical periods, contingencies, or stochastic realizations) with probability $\pi_s$, and $\rho$ is a penalty for unserved load (Value of Lost Load, VOLL, typically $1,000–$10,000 per MWh).
Operational constraints (per scenario $s$)
Bus balance at every bus $i$:
Existing line flow (DC approximation, standard):
Candidate line flow, disjunctive (the "big-M" formulation):
If $x_\ell = 1$, the equality (3) is enforced; if $x_\ell = 0$, the constraint becomes non-binding and flow is allowed to differ from angle difference (effectively the line doesn't exist).
Thermal limit conditional on build:
(For existing lines: $-F_\ell^{\max} \le f_{\ell,s} \le F_\ell^{\max}$.)
Generator limits, load-shed non-negativity:
Multi-year extension
The static formulation above fixes the build decision once; the multi-year dynamic TEP adds a year index: $x_{\ell, y}$ indicates whether line $\ell$ is built by year $y$, with accumulation $x_{\ell,y} \ge x_{\ell,y-1}$ (a built line stays built). Cost (1) is discounted: $\sum_y (1+r)^{-y} \ldots$. Multi-year TEP is what most real planners solve.
N−1 security constraint
Real-world TEP must produce a plan that remains operable after any single element outage. The N−1 formulation replicates constraints (2)–(5) once per contingency $k$: $f_{\ell,s,k}$, $\theta_{i,s,k}$, $u_{i,s,k}$. This explodes the LP size by a factor of $|K|$ (number of contingencies). Benders decomposition with contingency subproblems is the standard solution method.
Complexity & solution methods
- Direct MIP: branch-and-cut on the disjunctive formulation, up to $\sim$100 candidate lines.
- Benders decomposition: master problem chooses $x_\ell$; LP subproblems evaluate operational cost per scenario and cut back.
- Constructive heuristics: start with no new lines, iteratively add the one with largest marginal benefit per investment dollar until improvement saturates.
- Metaheuristics: genetic algorithms on the $x$ vector; popular in older TEP literature, less common now.
- Column-and-constraint generation: for robust TEP with budget-of-uncertainty sets.
Real-world data
Test networks & production practice
Garver's 6-bus system (1970)
The canonical TEP benchmark: 6 buses, 3 generators, 5 load buses, 15 candidate line additions (including parallel additions to existing corridors). Garver's original paper gives the line data in per-unit; his optimal solution adds 4 lines for a 20 pu-distance-weighted total investment. The interactive solver below uses this network.
IEEE 24-bus and 118-bus test systems
Larger academic benchmarks extending from operations (RTS-96) to planning. Widely used for N−1 secure TEP. MATPOWER provides ready network data.
NREL Interconnections Seam Study
The NREL Seams study (2016–2020) is a US-scale TEP applied to integrate Eastern, Western, and ERCOT interconnections. It found macro-scale transmission expansion essential for net-zero scenarios and produced publicly available network data used in several subsequent research papers.
Regional TEP publications
US ISOs publish TEP studies annually. PJM's Regional Transmission Expansion Plan (RTEP) and MISO's MTEP are the highest-profile examples. These use commercial solvers (PROMOD, PowrSym) plus engineering judgment.
Interactive TEP solver
Garver's 6-bus network · iterative constructive heuristic
Scenario parameters
6-bus network: existing + new
Solution interpretation
Where and why new lines are needed
The network diagram shows the set of existing lines (solid) and every candidate line (dashed if not selected, solid if selected by the solver). A congested existing line is shaded red; an available but unused candidate is thin-dashed grey. When the load multiplier is low, the existing network handles the flow and no new lines are built. When load scales up, certain corridors become bottlenecks and the solver adds the candidate line whose marginal benefit per dollar is highest — typically a line that relieves the most congested corridor or brings a stranded cheap generator into the load center.
The constructive heuristic used here (iteratively add the best remaining candidate) is not guaranteed globally optimal, but it captures most of the value and is standard in introductory TEP literature. Production TEP tools solve the full disjunctive MIP.
The investment-vs-operational-cost trade-off is the central insight. Building a line costs capital; not building it incurs ongoing operational cost (congestion rent, out-of-merit dispatch, eventually load shedding). The solver picks lines whose operational-cost savings over the planning horizon exceed their investment cost. Sensitivity to VOLL and load multiplier tells you how robust the plan is to alternative futures.
The dual of the line-limit constraint on a binding existing line is the shadow value of transmission capacity — the LMP spread from OPF. Persistent high shadow values across years are the economic justification for building. That dual is both an output of TEP and, conceptually, the reason TEP exists: the price signal from operations tells you where to invest.
Extensions & variants
Modern TEP research directions
Stochastic & robust TEP
Uncertain demand growth, renewable siting, fuel prices, and policy motivate scenario-based stochastic or adaptive-robust formulations. Solutions hedge against worst-case realizations; policy trade-off between expected cost and worst-case resilience.
Joint GEP-TEP
Co-optimize new generation and new transmission. Substantial savings versus sequential planning because generation siting can be chosen to minimize transmission need (or exploit cheap transmission).
AC-OPF-based TEP
Replaces the DC subproblem with AC power-flow constraints. Captures reactive power, voltage, and loss effects. Nonconvex MINLP; practical only for small networks unless convex relaxations (SOCP/SDP) are used.
Bi-level / MPEC TEP
When TEP anticipates a competitive market-clearing response, the lower level is the market-clearing LP and the upper level is the TNO's investment choice. MPEC with equilibrium constraints; used to study private vs social optima.
HVDC and multi-terminal networks
Long-distance transmission is increasingly HVDC (controllable, low-loss) rather than AC. TEP with HVDC has different decision variables (converter stations, point-to- point capacity) and different optimal topologies (radial vs mesh). Key for continent-scale renewable integration.
Distribution Network Expansion (DNEP)
TEP's sibling at the distribution voltage level: which new feeders to build, where to place transformers, how to upgrade under rising DER (solar rooftops, EV charging) penetration. Smaller networks but many more of them.