[Paper Review] Approximating Power Flow and Transmission Losses in Coordinated Capacity Expansion Problems

Fabian Neumann, Veit Hagenmeyer+1 more|arXiv (Cornell University)|Aug 26, 2020

Power System Optimization and Stability26 references4 citations

TL;DR

This paper evaluates linear approximations of power flow and transmission losses in coordinated capacity expansion models, showing that a linearised power flow with convex loss approximation using three tangents accurately captures physical behavior—reducing overestimation of grid expansion by 20% compared to lossless models—while enabling feasible post-optimisation AC power flow validation.

ABSTRACT

With rising shares of renewables and the need to properly assess trade-offs between transmission, storage and sectoral integration as balancing options, building a bridge between energy system models and detailed power flow studies becomes increasingly important, but is computationally challenging. W compare approximations for two nonlinear phenomena, power flow and transmission losses, in linear capacity expansion problems that co-optimise investments in generation, storage and transmission infrastructure. We evaluate different flow representations discussing differences in investment decisions, nodal prices, the deviation of optimised flows and losses from simulated AC power flows, and the computational performance. By using the open European power system model PyPSA-Eur we obtain detailed and reproducible results aiming at facilitating the selection of a suitable power flow model. Given the differences in complexity, the optimal choice depends on the application, the user's available computational resources, and the level of spatial detail considered. Although the commonly used transport model can already identify key features of a cost-efficient system while being computationally performant, deficiencies under high loading conditions arise due to the lack of a physical grid representation. Moreover, disregarding transmission losses overestimates optimal grid expansion by 20%. Adding a convex relaxation of quadratic losses with two or three tangents to the linearised power flow equations and accounting for changing line impedances as the network is reinforced suffices to represent power flows and losses adequately in design studies. We show that the obtained investment and dispatch decisions are then sufficiently physical to be used in more detailed nonlinear simulations of AC power flow in order to better assess their technical feasibility.

Motivation & Objective

• Address the computational challenge of integrating nonlinear AC power flow into large-scale linear capacity expansion models.
• Assess the trade-off between model fidelity, computational cost, and physical accuracy in transmission, generation, and storage expansion planning.
• Determine whether simplified linear flow models can produce investment decisions that are physically viable for subsequent nonlinear AC power flow analysis.
• Evaluate the impact of neglecting transmission losses and reactive power on investment and system cost outcomes.
• Provide guidance on selecting appropriate power flow approximations based on spatial resolution, computational resources, and application needs.

Proposed method

• Implement and compare five linear power flow models: transport model, transport with loss approximation, linearised power flow (lossless), linearised with loss approximation, and iterative linearised with loss approximation.
• Use PyPSA-Eur, an open-source, high-resolution European power system model, to simulate both linear optimisation and detailed AC power flow for validation.
• Approximate nonlinear quadratic transmission losses using a convex relaxation with two or three tangent lines to the quadratic loss curve.
• Update line impedances iteratively during the optimisation process to reflect capacity expansion, improving physical consistency.
• Validate results against full AC power flow simulations using Newton-Raphson method, including both distributed slack and voltage constraints.
• Assess model performance via computational time, solution accuracy, nodal price consistency, and deviation of flows and losses from AC simulations.

Experimental results

Research questions

• RQ1 How do different linear approximations of power flow and transmission losses affect investment decisions in coordinated generation, storage, and transmission expansion?
• RQ2 To what extent does neglecting transmission losses in linear models overestimate required grid reinforcement?
• RQ3 Can a convex relaxation of quadratic losses with three tangents accurately represent actual transmission losses in expansion planning?
• RQ4 How does iterative impedance updating improve the physical consistency of linearised power flow models?
• RQ5 Are investment and dispatch decisions from improved linear models physically feasible for subsequent AC power flow validation?

Key findings

• Disregarding transmission losses in the commonly used transport model overestimates optimal grid expansion by 20% compared to AC power flow simulations.
• Using a convex relaxation of quadratic losses with three tangents provides an accurate approximation of actual transmission losses while maintaining computational efficiency.
• Accounting for changing line impedances during reinforcement is essential; failing to do so leads to overestimation of losses and suboptimal investment decisions.
• The linearised power flow model with loss approximation produces investment and dispatch decisions that are physically viable and consistent enough to serve as inputs for detailed AC power flow simulations.
• The iterative linearised power flow with loss approximation shows the highest computational cost due to multiple solver iterations, but offers the best balance of accuracy and feasibility.
• Simpler models like the lossless transport model are insufficient for post-optimisation AC validation due to their lack of physical representation of losses and voltage behavior.