Robust Optimization for the Day-Ahead Scheduling of Cascaded Hydroelectric Systems

Uncertain electricity prices resulting from the re-structuring of electricity market have brought new opportu-nities and challenges for hydroelectric producers. This paper presents a data-driven robust optimization approach for the day-ahead scheduling of cascaded hydroelectric systems (CHS) with electricity price uncertainty. In this paper, a minimum volume enclosing ellipsoid (MVEE) is adopted to construct an ellipsoidal uncertainty set that fully identifies and exploits the historical characteristics data. A second-order conic optimization formulation of the robust counterpart is derived for efficient computation. A real-world case study is conducted to demonstrate the capability of the proposed optimization approach compared with the traditional robust optimization approach. Response to Reviewers We express our sincere thanks to the committee numbers and all the anonymous review-ers for their constructive comments on our manuscript. We have revised the manuscript based on the comments and suggestions. The detailed one-to-one responses are summarized below. The texts highlighted in blue are our response to the reviewers comments. Reviewer 1 Comment 1.1 - The innovation of this paper is not clear, as the modeling and conversion techniques have been well established. The authors are suggested to consider the case where CHS are distributed at locations with different electricity prices. Reply: Thanks for your comment. A point-by-point description on the major innovations of this paper is provided at the end of introduction section in page 2. Our contribution is to apply ellipsoidal uncertainty set and second-order cone optimization method to first capture the time-correlation characteristic of uncertain electricity prices for hydroelectric producers. Testing the performance of the proposed optimization technique in various cases is indeed important. Unfortunately, due to the page limitation, we are unable to supplement more numerical examples in this paper, so that it will be our future work. Comment 1.2 - The max operator in (8a) should be double-checked. Also, the authors should add more details (e.g., figures) regarding the piecewise linear approximation of the power output of hydroelectric units. Reply: Thanks for your comment. We have revised the typo in (8a) accordingly. Also, a figure (Figure 1 in page 3 of the revised manuscript) is provided to illustrate the piecewise linear approximation of the power output of hydroelectric units. Reviewer 2 Comment 2.1 - The notation is bad. For example,
$Q$
is used for both water flow rates and the symmetric positive definite matrix, while
$Q$
usually stands for reactive power. Reply: Thanks for your comment. In the revised version, we use
$R_{h_{p},t}$
instead of
$Q_{h_{p},t}$
to represent natural water inflow in case of confusion. Comment 2.2 - Eq. (1), it is said “The objective is to maximize the total profit.” However, only the revenue is considered, and the cost is not. Reply: Thanks for your comment. We do not consider cost terms in the objective function because the marginal generation cost of a hydroelectric unit is nearly 0.
$\ln$
this revised version, we supplement more explanations on this in case of confusion. Comment 2.3 - Subsection III-A is not clear. There is no motivation and intuition, only procedure. For example, what is the meaning of
$Q$
? And what is
$c$
? Reply: Thanks for your comment. A more intuitive explanation on how the values of
$Q$
and
$c$
is supplemented under Eq.(6) in page 3. Here
$c$
is the center of the ellipsoid, which can be regarded as the nominal value of
$\pi$
. Additionally, Q is a symmetric positive definite matrix. The eigenvalues of
$Q$
determine the lengths of symmetric axes, whose values restrict the upper and lower bounds of electricity prices. The eigenvectors of
$Q$
describe the rotation of the symmetric axes of the ellipsoid relative to the standard axes, whose values determine the correlations among the electricity prices in different time periods. Comment 2.4 - Subsection IV-A, “Historical electricity prices from 1/1/2021 to 10/31/2021 are collected from the website of CAISO,” why only consider 10 months of 2021, and the whole year of 2020? Reply: Thanks for your comment. We consider the recent 10 months because the samples whose dates are closer to the scheduling day are more suitable to construct uncertainty set. Additionally, having more historical samples do not always mean that better solution can be derived, so it is not necessary to include such a lot of historical samples in the numerical experiments. Comment 2.5 - What is the computational efficiency of the new approach as compared with the existing ones (e.g., solving time) ? Reply: Thanks for your comment. We supplement some descriptions on the computational time of the proposed model in the revised manuscript (page 5). The average computational time for the proposed second-order conic optimization model under ellipsoidal set is 4.88 seconds, and the average computational time for the traditional linear optimization model under polyhedral set is 4.32 seconds. These shows that considering ellipsoidal set does not significantly affect the computational efficiency in this case study. Comment 2.6 - The third contribution “A comprehensive statistical analysis between tradition polyhedral uncertainty set and the proposed ellipsoidal uncertainty set is conducted based on the real-world data from California ISO (CAISO).” Only a few cases were tested, which cannot be stated as “comprehensive statistical analysis.” Reply: Thanks for your comment. We modify the description on the third contribution by removing the word “comprehensive” according to your comment (page 2). Reviewer 3 In this paper, the authors proposed a robust optimization method to schedule cascaded hydroelectric systems considering the price uncertainty of the day-ahead market. The pro-posed method is tested using real-world data and compared with the traditional optimization methods to show its advantages. The paper is well prepared. The authors should be commended for their nice work. Reply: We would like to express our sincere gratitude for your appreciation to our work! Reviewer 4 This paper develops a robust optimization model to optimize the DA scheduling of cascaded hydroelectric systems in the presence of electricity price uncertainties. The authors use a data driven method to develop an ellipsoidal uncertainty set for uncertain prices. The bi-level robust model is converted to a single level second-order cone programing problem. A numerical example is used to demonstrate the effectiveness of the proposed model. The paper is well written. One minor comment is the following: could the authors double check if the second-order conic problem (12a) should include constraints (11b)-(11d). Reply: Thank you so much for your summary on our paper! We have revised the typo in (12a) according to your comment.