Master's Thesis

Financial Optimization Models

Final Thesis 5.3 MB Appendix 1.08 MB

Author of thesis: Rishika Nayana Patil

Acad. year: 2025/2026

Supervisor: RNDr. Pavel Popela, Ph.D.

Reviewer: Ing. Vít Procházka, Ph.D.

Abstract:

(Medium term planning of a cascade hydroelectric plant – Multistage
Stochastic Optimization)
This thesis investigates the application of multistage stochastic optimization to maximize the revenue generated by a cascade hydroelectric plant, with a specific focus on the hydroelectric plants of the Vltava cascade. The study utilizes real data: natural water inflow into reservoirs, electricity market price and hydroelectric plant technical specifications, to develop a 16 week multistage evaluation of the expected revenue. The model involves simulating data and a scenario tree, and optimizing for expected revenue. The optimization is implemented using the HiGHS solver on Python, which provides an understanding of the expected water movement in the river and the range of revenue generated at each time step. The results demonstrate the effectiveness of mathematical programming in medium to long term planning
of water storage and utilization to generate electricity and support data informed decision making in hydro energy planning.

Keywords:

multistage stochastic optimization, cascade hydroelectric plant, scenario tree generation, Wasserstein distance, AR(1) process, revenue optimization, Vltava cascade, nonanticipativity, linear programming

Date of defence

16.06.2026

Result of the defence

Defended (thesis was successfully defended)

znamkaAznamka

Grading

A

Process of defence

The student presented her work on the topic “Financial Optimization Models”. The supervisor was present in person and read his review. The Secretary then read the opponent’s review. After that, the student responded suitably to the opponent’s questions.

Language of thesis

English

Faculty

Department

Study programme

Applied and Interdisciplinary Mathematics (N-AIM-A)

Composition of Committee

doc. Ing. Luděk Nechvátal, Ph.D. (předseda)
prof. RNDr. Josef Šlapal, CSc. (místopředseda)
Mgr. Jitka Zatočilová, Ph.D. (člen)
doc. Ing. Jiří Šremr, Ph.D. (člen)
prof. RNDr. Miloslav Druckmüller, CSc. (člen)
Prof. Raffaele D'Ambrosio (člen)

Supervisor’s report
RNDr. Pavel Popela, Ph.D.

The submitted thesis fully meets and in several aspects rather surpasses the specified requirements. In the course of our discussions, we gradually refined and specified the topic so that it more clearly focused on optimization in the energy sector from an economic–financial perspective, which proved to be a very natural and, in retrospect, fortunate shift. The student successfully  combines theoretical knowledge with a realistic case study of the Vltava cascade. All declared goals ranging from the formulation of stochastic optimization models, through their implementation, to the interpretation of results have been fulfilled in a coherent and meaningful way.
It is important to emphasize that the author worked independently and with a quiet persistence that I appreciated more and more as the work progressed. She showed an understanding of the topic in great depth. The working process was systematic, thoughtful and, at times, pleasantly self-driven. She also attended the related seminar in Prague, which she used not merely as a formality but as an opportunity to broaden her view of the problem. Especially, a valuable part of her work was the intensive communication with Czech specialists from practice regarding data acquisition and structural aspects of the model. This was, as one might expect, time‑consuming, yet very successful and beneficial for the overall quality of the thesis.
The choice of multistage stochastic programming as the central methodological framework is fully adequate to the problem of hydroelectric cascade management under uncertainty. The implementation in Python with the HiGHS solver demonstrates practical competence and a rare ability to move from theory to functioning code. I appreciate the construction of a  scenario tree and its connection with real hydrological and market data, which provides a solid basis for subsequent optimization experiments. The originality of the work lies less in proposing entirely new theory and more in the thoughtful synthesis and application of established methods in a coherent framework, which is fully appropriate. According to the Theses.cz similarity report, the overall similarity is low and indicates standard and acceptable use of sources.  
The student demonstrates a good ability to interpret the results of the implemented model and to relate them back to the studied problem. The discussions are clear and logically structured. The identified limitations and proposals for further work show a realistic awareness of the model’s scope. The results are, in principle, applicable both in theoretical development and in practice, particularly in medium-term planning of hydroelectric systems. The use of real data from the Vltava cascade enhances the practical relevance. At the same time, certain simplifying assumptions slightly limit direct applicability. The structure of the thesis is logical and follows the standard progression from introduction and theory to data, modelling and results. Chapters are arranged in a clear and coherent manner, and transitions between sections are mostly smooth.
Minor formal shortcomings appear sporadically, however, nonanticipativity deserved to be also presented with the model not only in general and in the Pytthon code. The graphical and typographical quality of the thesis is overall very good. Figures and tables are generally clear, informative, and appropriately referenced. The text reads fluently and without unnecessary ornamentation. Some linguistic imperfections occur. In a certain sense, they even lend the text a human quality that feels closer to real scientific writing than overly polished formulations sometimes do. The student works with the literature in a correct and adequate manner. The selection of sources reflects the theoretical background required for stochastic programming and its applications, including classic monographs and more specialized materials. Citations are used appropriately, and the bibliography is relevant to the topic.
The thesis represents a well-prepared and thoughtfully executed piece of work that combines theoretical understanding with practical implementation. It was prepared by a student who, throughout her studies, consistently demonstrated independence, diligence and a quiet but very reliable working style. She approached the topic with a certain modest determination, always prepared, often already a step ahead, and her willingness to study more specialized sources on her own deserves explicit appreciation.
I recommend the submitted master’s thesis for defence and evaluate it overall as excellent, so A.
Evaluation criteria Grade
Splnění požadavků a cílů zadání A
Postup a rozsah řešení, adekvátnost použitých metod A
Vlastní přínos a originalita A
Schopnost interpretovat dosažené výsledky a vyvozovat z nich závěry A
Využitelnost výsledků v praxi nebo teorii B
Logické uspořádání práce a formální náležitosti B
Grafická, stylistická úprava a pravopis A
Práce s literaturou včetně citací A
Samostatnost studenta při zpracování tématu A

Grade proposed by supervisor: A

Reviewer’s report
Ing. Vít Procházka, Ph.D.

This thesis develops and implements a multistage stochastic optimization model for revenue maximization of the Vltava cascade hydropower system in the Czech Republic. The author formulates a risk-neutral multistage linear program over a 16-stage scenario tree and solves the resulting extensive-form LP using HiGHS via Pyomo. The topic is original and technically demanding. The presentation of the thesis is clear, the idea and theoretical background are well explained, and the results are nicely presented. I appreciate the rich discussion of limitations and potential further improvements, which gives the impression that the author thought deeply about the model and its behavior.

I identified some errors and spots for potential improvement that I would like to point out:

- Section 6.1 indexes all decision variables by time step t only (and i), which makes the written model appear deterministic. A correct extensive-form multistage stochastic linear program must index all decision variables over scenario tree nodes. Nonanticipativity, which the theoretical chapter correctly identifies as the central structural requirement of multistage programming, is never stated as a constraint in the formal model. It is worth noting that the attached implementation is correct — decision variables are properly indexed over tree nodes and nonanticipativity is enforced structurally — so this is a presentational gap in the write-up rather than an error in the computation.

- The revenue expression in Sections 6.1.5 and 6.1.7 should have "+" instead of "−", given that x^i_t is negative for k^i < 0. Again, the code handles this correctly by using absolute values of k[a] for pump arcs together with an explicit minus sign, so the reported numerical results are unaffected.

- The final model formulation (6.1.8) should explicitly state the sets over which each constraint applies, even if these may seem straightforward in context.

- The conclusions and applicability of the results to real-world decision-making are limited by the end-of-horizon modeling and the absence of in-sample and out-of-sample stability tests for the input scenarios. The size and composition of the scenario tree are not properly justified. I include these points as questions for the defense, as they represent the most promising directions for further improvement.

I recommend that the submitted thesis be accepted as a Master's thesis. I propose a grade of B; however, if the author demonstrates a very good understanding of the topic and gives convincing answers to my and the committee's questions during the defense, I could also support a higher grade, as the identified issues of the thesis were mainly related to the formal description of the model. The main outcome of the thesis - the functional code with an optimization model and its results - was correct.
Evaluation criteria Grade
Splnění požadavků a cílů zadání A
Postup a rozsah řešení, adekvátnost použitých metod B
Vlastní přínos a originalita A
Schopnost interpretovat dosaž. výsledky a vyvozovat z nich závěry A
Využitelnost výsledků v praxi nebo teorii A
Logické uspořádání práce a formální náležitosti C
Grafická, stylistická úprava a pravopis B
Práce s literaturou včetně citací B
Topics for thesis defence:
  1. The scenario tree uses 256 leaf paths for a six-dimensional stochastic process. How would you test whether this number of scenarios is sufficient to produce a stable solution? The end-of-horizon constraint requires all reservoirs to hold at least 70% of their operational capacity at week 16, but no value is assigned to water stored above this floor, meaning the optimizer has no incentive to conserve water beyond what the constraint forces. Do you see a better way to handle the terminal stage? For example, how would you construct a terminal value function for the stored water, and what data or additional modeling would be required to calibrate it?

Grade proposed by reviewer: B

Responsibility: Mgr. et Mgr. Hana Odstrčilová