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Modeling of Network Constraints in Large-scale Capacity Expansion Optimization of Power Grids
Yu Weng, Lara Booth, Priya L. Donti, Ruaridh Macdonald
TL;DR
Large-scale capacity expansion models often omit nonlinear, non-convex ACOPF constraints, limiting analysis of how network physics affects investment decisions. The paper extends GenX with fixed-point theorem-based network constraints and compares them with transport and full ACOPF formulations in ISO New England. The fixed-point approach closely reproduces ACOPF planning patterns while retaining near-transport model dimensions and much lower computational cost.
Problem
CEM simplifications improve tractability but limit understanding of how AC network physics affects generation, storage, and transmission investment decisions.
Method
The paper integrates full ACOPF into GenX and develops fixed-point theorem-based network constraints that embed AC-related considerations in the transport-based MILP.
Results
FX constraints produce planning decisions closer to ACOPF than transport, with 10% median absolute difference in individual generator capacities for the tightest bounds and 5–20% runtime increases.
Takeaways & Limitations
Fixed-point constraints offer an accuracy–cost trade-off for incorporating ACOPF effects into large-scale capacity planning.
Abstract
from arXiv · showhide
Capacity expansion modeling plays a critical role in optimizing the deployment of new generation, storage, and transmission, typically at national and regional levels. To support long-term planning, these models consider a large set of energy technologies and policies, along with decades of weather and demand data. Realistic capacity expansion models thus become high-dimensional optimization problems, with hundreds of millions of variables and constraints, which are challenging to solve. A common strategy to address this complexity is to omit non-linear, non-convex AC optimal power flow (ACOPF) constraints and instead use linearized power balance equations or transport formulations. While these simplifications improve tractability, they limit our understanding of how power flow and the physical properties of power networks impact investment decisions across generation, storage, and transmission infrastructure. This paper addresses this gap by extending the GenX capacity expansion model to incorporate fixed point theorem-based network constraints. These embed ACOPF-based considerations while maintaining the tractability of the planning model, nearly preserving the dimensionality of the transport formulation and incurring only modest runtime increases. This approach is much cheaper than embedding ACOPF directly, making it appropriate for large-scale capacity planning problems. We compare our approach to the original transport-based GenX model as well as a non-linear, non-convex version that incorporates the full ACOPF constraints, for a case study of the ISO New England grid.
I. INTRODUCTION
Capacity expansion models simplify detailed power-system physics to remain tractable, leaving uncertainty about how AC network constraints affect investment decisions. This work extends GenX with ACOPF and fixed-point-based constraints to study that trade-off at scale.
- Motivation: CEMs optimize technologies, policies, and high-resolution operational data, creating high-dimensional problems that are generally difficult to solve.Models reduce computational burden through temporal or spatial aggregation, linearization, resource aggregation, and decomposition, at the cost of fidelity.
- Research gap: Most large-scale CEMs omit nonlinear, non-convex ACOPF constraints and instead use linearized power balance or transport formulations.Prior studies found that more advanced transmission representations changed planning results, but did not compare directly with full ACOPF CEMs.
- Contribution: The paper integrates AC power-flow constraints into GenX to assess how AC grid constraints influence generation, storage, and transmission investment decisions.The stated objective is a more physically faithful assessment within a large-scale capacity modeling framework.
- Findings: ACOPF-based planning installs additional generation capacity and distributes generation more evenly across nodes.The reported drivers are reactive power demand, reduced generator active-power dispatch capacity, reduced network transfer capacity, and greater losses.
- Contribution: Fixed-point theorem-based network constraints reproduce ACOPF-like behavior in linear CEMs while preserving computational tractability.The approach retains the transport formulation’s decision variables and is validated against GenX with full nonlinear, non-convex AC power flow.
III. FULL AC POWER FLOW AND FEASIBILITY CONSTRAINTS IN CEMS
The paper embeds full AC optimal power flow constraints into GenX to enforce physically detailed feasibility of dispatch, transfers, and voltages. This improves fidelity but produces a non-convex mixed-integer nonlinear model with substantially higher computational cost.
- ACOPF integration: ACOPF constraints govern generator dispatch, power transfers, and voltage levels across zones and time periods in GenX.The formulation is adapted from Power Grid Lib for multiple time steps.
- ACOPF formulation: The AC formulation represents complex voltages, generation, bidirectional line flows, voltage bounds, admittances, and line charging susceptance.These quantities are defined over time periods, zones, generators, and transmission lines.
- ACOPF integration: AC active generation variables are linked to GenX dispatch, while zonal balances include incident-line flows and shunt conductance.Transmission losses use forward and reverse active flows, and the transport model’s fixed-fraction loss term is removed to avoid double-counting.
- ACOPF formulation: The model constrains apparent power flow by available transmission capacity while retaining GenX’s linear operational cost objective.No relaxation is applied to the ACOPF constraints.
IV. NETWORK FEASIBILITY CONSTRAINTS IN CEMS
The proposed fixed-point network constraints add AC-related feasibility considerations to transport-based GenX without adding decision variables. They preserve key AC physical characteristics while keeping the planning model tractable.
- FX formulation: FX constraints are embedded directly into the original MILP GenX transport formulation using the same decision variables and constraints.Only additional FX constraints are added to the transport formulation.
- Evaluation: The solve-time comparison varies spatial zones and representative weeks, with all runs subject to a 3 hour time limit.Cases use 6 or 12 zones and representative-week counts of 1, 2, 3, and multiples of 4 through 52.
- Purpose: The FX approach is designed to retain key physical characteristics of AC power systems while maintaining computational tractability.The section derives a CEM-adapted certificate and describes how to embed it in a capacity expansion model.
A. Fixed-point theorem and derived certificate
The FX feasibility approach uses fixed-point certificates to characterize admissible AC power-flow injections. Kantorovich-based conditions provide a unique stable solution and Newton–Raphson convergence, but applying them requires reformulating the complex power-flow equations.
- Feasibility set: The FX feasibility set defines power injections satisfying steady-state AC network stability and transmission-line capacity limits.It is computed using an advanced fixed-point theorem-based certificate.
- Fixed-point theorem: Fixed-point theorems assess solvability by seeking a point x satisfying x = F(x), and can therefore assess existence of AC power-flow solutions.The mapping formulation provides a lens for analyzing whether the equations have solutions.
- Fixed-point theorem: Kantorovich’s fixed-point theorem specifies conditions on an initial point x0 and the Jacobian J(x) of F(x).It is closely related to Newton–Raphson and is selected for integration with AC power-flow equations.
- Certificate guarantee: When the conditions hold, the theorem guarantees a unique solution within the admissible region and convergence of Newton–Raphson initialized at x0.The certificate uses constants and norm bounds to establish these guarantees.
- Technical challenge: Applying the theorem to power systems requires reformulating S = V I* because complex conjugation prevents the standard relation from being analytic.The reformulation must preserve physical power-flow properties and avoid a conservative admissible-injection region.
2) Fixed-point theorem-based certificate for power systems:
The paper extends a fixed-point certificate for AC power flow into capacity expansion modeling, adapting it for aggregated, active-power-only formulations. The resulting constraints define admissible injections while accounting for network stability, transmission limits, and reference-point margins.
- Fixed-point certificate: The certificate provides sufficient conditions for steady-state stability and convergence to a unique AC power-flow equilibrium around a nominal operating point.It applies to injections satisfying the certificate inequalities and assumes a well-defined nominal point.
- Assumptions: The certificate’s guarantees depend on the nominal point and assume a static network topology and impedance matrix.A poorly chosen nominal point can make the condition more conservative or reduce generalization to substantially different scenarios.
- CEM adaptations: The CEM adaptation addresses zonal aggregation, mixed generator-load zones, engineering-unit calculations, and the absence of fixed bus types.These differences require modified certificate conditions and rescaled variables for capacity expansion models.
- CEM adaptations: The formulation converts complex apparent-power constraints into active-power constraints using reference-solution power factors and a correction factor for aggregation.The candidate zonal injections are constrained through a margin parameter computed from a reference ACOPF-CEM solution.
- Reference-point margin: The margin parameter λ rescales the admissible region relative to benchmark line-flow limits and remains fixed during subsequent FX-constrained optimization.The construction uses apparent-power magnitudes, power factors, line capacities, and reference line flows.
- Bound variants: Tight, medium, and loose bounds respectively use hourly reference flows, maximum reference flows, and line capacities, trading conservatism against flexibility.Looser bounds admit solutions closer to the transport-model optimum, but still restrict deviations from the nominal point.
C. Integrating FX network feasibility constraints into CEMs
FX network feasibility constraints are generated from a shorter-horizon ACOPF reference and embedded as linear constraints in the transport-based CEM. The formulation constrains active power without explicitly modeling reactive power.
- Integration workflow: The FX-CEM workflow computes an ACOPF reference, derives the certificate, embeds its linear constraints in the transport CEM, and solves the constrained planning problem.The reference scenario uses the same topology and typically a shorter time horizon.
- Model scope: FX-CEM does not explicitly model reactive power, instead constraining active-power variables to values intended to imply admissible reactive-power operation.This is an approximation rather than a direct reactive-power representation.
V. NUMERICAL EXPERIMENTS AND VALIDATION
The numerical validation uses an aggregated three-zone ISO New England case with twelve representative weeks and compares transport, full ACOPF, and three FX-bound formulations.
- Case study: The ISO New England case aggregates Massachusetts, Connecticut, and Maine and uses one year of hourly active demand with synthesized reactive demand.Twelve one-week representative periods were selected by k-means clustering to make the ACOPF formulation tractable.
- Computational setup: The experiments were implemented in Julia and solved with IPOPT v3.14.19 using the MA86 linear solver, MC64 scaling, and an adaptive µ-strategy.All runs used a MacBook Pro with a 2.8 GHz Intel Core i7 processor and 16 GB memory.
- Compared formulations: Five transmission formulations are compared: transport, ACOPF, and FX-Tight, FX-Medium, and FX-Loose.The FX constraints are calculated from an ACOPF reference solution obtained using one representative week.
- FX bounds: Figure 2 visualizes tight, medium, and loose FX feasibility-bound ranges for twelve representative weeks of one year.Loose-bound parameters are rescaled to one-sixth of their original values for visualization.
A. Generation of AC-Specific Model Data
The AC-specific data preparation synthesizes reactive demand and estimates missing AC line parameters for the three-zone case. These inputs support ACOPF and FX feasibility calculations over twelve representative weeks.
- Reactive demand: Reactive demand is synthesized from active-load time series using a uniformly sampled power factor between 0.89 and 0.91.Technology-specific reactive limits use power factors of 0.90 for thermal generators, 0.98 for storage, and 0.99 for variable renewables.
- Representative periods: Figure 2 reports FX feasibility-bound ranges for twelve representative weeks selected from one year.The figure presents the bound ranges used for the tight, medium, and loose FX variants.
- Network parameters: AC line resistance, reactance, and charging susceptance are estimated from a standard 380 kV overhead-line type and scaled by line length.The model uses 60 Hz frequency, 345 kV base voltages, and zonal voltage magnitudes between 310 and 380 kV.
B. Investment Decisions in the Planning Problem
ACOPF-based planning changes investment outcomes by requiring more capacity and shifting generation locally, while constraining active power transfers through reactive-power effects.
- Investment differences: The transport formulation underestimates required new capacity, with a 23% median absolute difference in individual capacities relative to ACOPF.The difference reflects reactive-power requirements and reduced active-power transfer capability.
- Investment differences: Figure 3 compares installed generator and storage capacities across five scenarios and their Euclidean distances from the ACOPF scenario.Panel A shows capacities; panel B shows distances for the labeled scenarios.
- Network-flow effects: ACOPF limits active power flow between Massachusetts and Connecticut to 3738 MW, compared with the 5730 MW transmission limit reached under transport.The corresponding ACOPF apparent-power maximum is approximately 5277 MW.
- Network-flow effects: Reduced active-power transfer capacity under ACOPF requires more local generation.Reactive power flow and network losses reduce the ability to transmit active power.
- Network-flow effects: Figure 4 compares active power flow between Massachusetts and Connecticut under different formulations.The comparison highlights how network constraints alter interzonal transfer patterns.
2) Planning with FX constraints:
Fixed-point network constraints produce planning outcomes closer to ACOPF than transport formulations while adding few constraints and only modest computational overhead.
- Planning outcomes: FX-Tight yields a 10% median absolute difference in individual generator capacities from ACOPF, compared with 16% for FX-Medium and 30% for FX-Loose.The loose FX bound performs worse than the transport formulation on this measure.
- Planning outcomes: Total system costs are similar across formulations, with ACOPF highest, transport lowest, and FX cases ordered by bound tightness.Figure 5 reports each case’s total cost and its percentage difference from ACOPF.
- Physical fidelity: FX constraints capture AC-physics effects on network congestion without explicitly representing reactive power.FX-CEM active-power flows are similar to ACOPF flows, although explicit reactive-power representation may improve accuracy.
- Planning outcomes: Figure 5 compares total costs across cases and annotates each case with its percentage difference from the ACOPF result.The figure supports cost comparisons among transport, ACOPF, and FX formulations.
- Computational costs: FX adds 3% of the original model’s constraints and increases runtime by 5–20%, whereas ACOPF increases variables and constraints by 50% and solve time by 4×.The transport formulation remains the smallest and fastest baseline.
- Computational costs: The FX formulation does not introduce new variables and retains computational tractability for large-scale capacity expansion modeling.Its absolute-value expressions were linearized without loss of generality in subsequent GenX versions, producing a MILP.
D. Robustness of Computed Parameters
The study tests whether FX network constraints calibrated from different demand cases remain useful for investment planning. Both FX formulations produce investment results closer to the ACOPF reference than the transport formulation, indicating robustness to out-of-sample demand profiles.
- D. Robustness of Computed Parameters: The second case study uses lower fuel prices, while the third and fourth also alter demand timeseries.The third case has lower peak demand during the first case’s peak hours; the fourth has uniformly higher load.
- D. Robustness of Computed Parameters: The second case study is solved with transport, ACOPF, and two FX formulations calibrated from the third and fourth cases.The FX formulations use tight bounds and reference points from those separate case studies.
- D. Robustness of Computed Parameters: Both FX formulations produce investment results closer to the ACOPF reference than the transport formulation.Figure 7 compares installed-capacity decisions across all four formulations.
- D. Robustness of Computed Parameters: FX parameters calibrated on one dataset generalize to out-of-sample demand profiles on the same network.The paper identifies this robustness as valuable for planning under uncertainty and very long time periods.