To address this issue, we propose three new mixed-integer linear programming UC models capable of modeling adaptable power output trajectories during multi-period SU and SD procedures of thermal generators. We denote this feature as “operational flexibility”. The first model serves as a baseline for modeling this operational flexibility and allows generator operators to define various power output domains (which we denote as “paths”) within which appropriate SU/SD trajectories lie. The second model aims to more accurately capture costs associated with these adaptable startup and shutdown transitions by allowing generator operators to define a piecewise-linear relationship between various power output trajectories and their associated costs. The third model enables flexibility in both power output and duration of startup and shutdown transitions. This allows generator operators to capture similar levels of detail (e.g. both long and short SU/SD trajectories) using fewer paths.
Using a series of generated instances, we then evaluate the efficacy of our models in two respects: Solution time and solution quality. For each, we compare the results of our models to the results of two of the most similarly-capable models (e.g. capable of modeling multi-period SU/SD trajectories) found in the literature (i.e. [4,5]).
With respect to solution time, commercial UC optimizations must generally be solved within a fixed amount of time in order to not disrupt market proceedings. With the 1-hour limit we imposed on each models’ optimization runs, our models were able to solve more than 90% of the instances that the literature models could also solve within the timeframe. With this in mind, we conclude that, while not being the fastest UC models available, our models are at least acceptably fast. We also maintain that the true value of our models comes from their ability to produce better quality solutions.
With respect to solution quality, we seek to compare the execution of the commitment schedules produced by our models with those of existing models. However, the actual execution of the electricity market— how the UC optimization’s solution is actually used in practice— is fairly complex. Thus, gauging the true quality of our solutions versus those of existing models on real-life, physical equipment is very difficult. While this is an ongoing focus of our future work, for the purpose of this presentation, we take the optimal objective function value of each model as a metric of its overall solution quality. With this in mind, we present a probability density function based on our generated instances. This function outlines the likelihood of experiencing a given fractional improvement of the optimal objective when switching from existing models to our models. We observe a maximum fractional reduction of 30.2% and a median fractional reduction between 0.133% and 0.427% depending on the model. While these numbers may seem small, considering the vastness of the transactions occurring each day throughout an electrical grid, even such small improvements are quite significant.
We also acknowledge the infantile nature of considering such operational flexibility in standard UC contexts. With the results we present, we believe more research into this notion, both from a grid operator perspective and from a generator operator/manufacturer perspective is likely to be rewarding. We argue that these models are a pivotal tool to quantify the benefit of considering operational flexibility during startup and shutdown.
References
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[2] L. Montero, A. Bello, J. Reneses, and M. Rodriguez, “A Computationally Efficient Formulation to Accurately Represent Start-Up Costs in the Medium-Term Unit Commitment Problem.” IEEE Transactions on Power Systems, vol. 38, no. 6, pp. 5623-5634, 2023.
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