(12g) Two-Stage Stochastic Generalized Disjunctive Programming (GDP) Model and Algorithm for Proactive Planning and Operations of Resilient Power Systems Under Disruptions

Reliability is the ability of power systems to satisfy power demands, and resilience is the ability of power systems to quickly revert to their normal operating conditions after a disruption [1]. While resilience is related to low probability and high impact (LPHI) events such as natural disasters and/or extreme weather conditions, reliability is concerned with high probability and low impact (HPLI) events such as equipment failures. Power system reliability can be enhanced by installing sufficient capacity, and highly reliable power systems can constantly produce power and satisfy power demand without interruption. On the other hand, power system resilience analyzes the recovery process after a power outage. Resilience studies take preventive measures to reduce the probability of power outages and suggest ways to restore damaged power systems quickly [2]. Establishing resilient power systems is becoming increasingly important in modern energy systems with a higher share of variable generation such as wind turbines and solar panels.

In this work, we propose a new optimization model for the expansion planning of resilient power transmission systems. A two-stage stochastic Generalized Disjunctive programming (GDP) model [3] that aims to proactively plan for the capacity of transmission and re-dispatch power generation systems has been proposed, mitigating potential electricity supply disruptions caused by extreme weather events. The objective function of the proposed model is to minimize the total cost, which includes investment (for new installation and hardening), fixed/variable operating costs, and resilience-related penalties such as load shedding and curtailment penalties. While the first stage decisions (also known as here-and-now decisions) include installing new transmission lines and hardening existing lines, the second stage decisions (also known as wait-and-see) involve re-optimizing power outputs and flows in extreme weather-induced failures. The following programming assumptions to develop the model are also applied: a) knowledge of which lines will be attacked and when, b) the transmission lines can withstand the attack if the lines’ resilience is above the threshold, and c) hardening cost is limited by hardening budget.

Lagrangian decomposition to address a computational challenge for large-scale problems has been proposed and implemented. The model was tested with the 5-year planning problems: a) 3-buses problem, b) 10-buses problem, and c) 50-buses problem. All case studies were initially evaluated using two different reformulations (i.e., Big-M and Hull relaxation) applicable in Pyomo.GDP [4], and the decomposition algorithm was applied to solve a large-scale problem. It was proved that Big-M reformulation takes less computational time than others because it has fewer constraints and continuous variables. It was also verified that increasing the number of hardened lines increases investment costs, and load-shedding and curtailment penalties decrease, resulting in a decrease in total costs.

Keywords: Resilience, Transmission expansion, Hardening, Stochastic programming

Disclaimer This project was funded by the Department of Energy, National Energy Technology Laboratory an agency of the United States Government, through a support contract. Neither the United States Government nor any agency thereof, nor any of their employees, nor the support contractor, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Acknowledgments This work was conducted as part of the Institute for the Design of Advanced Energy Systems (IDAES) with support from the U.S. Department of Energy’s Office of Fossil Energy and Carbon Management through the Simulation-based Engineering Research Program.

References

[1] Cho, S., Li, C., Grossmann, I.E., “Recent advances and challenges in optimization models for expansion planning of power systems and reliability optimization”, Computers and Chemical Engineering, 165, 107924 (2022).

[2] Gacitua, L., Gallegos, P., Henriquez-Auba, R., Lorca, Á., Negrete-Pincetic, M., Olivares, D., Valenzuela, A., Wenzel, G., “A comprehensive review on expansion planning: Models and tools for energy policy analysis”, Renewable and Sustainable Energy Reviews, 98, 346-360 (2018).

[3] Grossmann, I.E., and Trespalacios, F., “Systematic Modeling of Discrete-Continuous Optimization Models through Generalized Disjunctive Programming,” AIChE Journal, 59, 3276-3295 (2013).

[4] Chen, Q., Johnson, E. S., Bernal, D. E., Valentin, R., Kale, S., Bates, J., Siirola, J. D. and Grossmann, I. E., “Pyomo.GDP: an ecosystem for logic based modeling and optimization development”, Optimization and Engineering, 23, 607-642 (2022).