Эффективные подходы на основе данных к задачам стохастического оптимального распределения потоков электроэнергии/Efficient Data-Driven Approaches in Stochastic Optimal Power Flow тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Лукашевич Александр Леонидович

• Лукашевич Александр Леонидович
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Лукашевич Александр Леонидович

Contents

Page

Introduction

Chapter 1. Power system

1.1 Alternating Current Basics

1.1.1 Phasors

1.1.2 Electrical Power

1.2 Power Grid

1.2.1 Topological Model of a Power System

1.2.2 Bus Classification

1.2.3 Power transmission lines

1.2.4 Kirchhoff Current law and Admittance Matrix

1.3 Power Flow Equations

Chapter 2. Optimal Power Flow - OPF

2.1 Introduction

2.2 Optimization problem

2.3 Constraints

2.3.1 Equality constraints

2.3.2 Inequality constraints

2.4 DC OPF Approximation

2.5 Dynamic DC-OPF and Automated Generation Control

Chapter 3. Power Grid Reliability Estimation via Adaptive

Importance Sampling

3.1 Introduction

3.2 Background and Problem Setup

3.3 Overload Probability Estimation

3.3.1 A Single Constraint Case

3.3.2 Multiple Constraints Case

3.3.3 Convexity of Importance Sampling Variance

3.3.4 Numerical Method

Page

3.4 Empirical Study

3.4.1 Algorithms and implementation details

3.4.2 Test cases and numerical results

3.5 Conclusion

Chapter 4. Importance Sampling Approach to Chance

Constrained DC-OPF

4.1 Introduction

4.2 Background and Problem Setup

4.2.1 Notation

4.2.2 Problem Setup

4.2.3 Scenario Approach

4.3 Algorithm

4.3.1 Idea and Sketch

4.3.2 Inner Approximation

4.3.3 Redundant Scenarios

4.3.4 Importance Sampling

4.3.5 Scenario Approximation with Importance Sampling

4.4 Empirical Study

4.4.1 Implementation details

4.4.2 Test Cases and Numerical Results

4.5 Conclusion

Chapter 5. A-Priori Reduction of Scenario Approximation for

Dynamic DC-OPF

5.1 Introduction

5.2 Background and Problem Setup

5.2.1 DC Optimal Power Flow

5.2.2 Source of uncertainty and AGC

5.3 Optimal multi-stage control under uncertainty

5.4 Chance constraint multi-stage control

5.4.1 Chance constrained optimization

5.4.2 Scenario approximation of chance constrained control

5.5 A-priori scenario redundancy

Page

5.5.1 Redundant scenarios

5.5.2 SA Solution Guarantees and Dataset Complexity

5.6 Empirical Study

5.6.1 Algorithms and Implementation Details

5.6.2 Test Cases and Numerical Results

5.7 Conclusion

Conclusions

List of symbols and abbreviations

Bibliography

List of Figures

List of Tables

Введение диссертации (часть автореферата) на тему «Эффективные подходы на основе данных к задачам стохастического оптимального распределения потоков электроэнергии/Efficient Data-Driven Approaches in Stochastic Optimal Power Flow»

Introduction

Background. A general way to model and quantize uncertainty in an optimization problem is the Chance Constraints [1]. Such kind of constraints allow one to properly incorporate uncertainty into optimization problems and define probability level at which one allows the system to violate constraints. The allowance for original deterministic constraints violation is important to avoid excessive solution conservatism. For example, finding a solution that is robust against each possible realization of uncertainty with unbounded support would result in an empty feasibility set. However, in most cases, chance constraints do not admit a closed-form expression, thus, making it impossible to use them in numerical solvers. These constraints are typically approximated using upper bounds on probability, e.g., Bernstein approximation, [2] or using data, i.e., constructing a data-driven approximation. Further, we focus only on the latter type of approximations.

Chance constrained optimization originated in 1958 by Charnes [3] and since then found a wide range of applications. For example, in economics [4], control theory [5], chemical processes [6] and in machine learning [7; 8]. Chance constraints are generally do not admit a closed-form expression. To this end, several approaches were developed to construct various approximations chance constraints. These approaches further evolved into major research and application fields. Among them are Abmiguous Chance Constraints [9; 10], Robust Optimization (RO) based Methods [11], Sample Average Approximation (SAA) [12; 13] and Scenario Approach (Approximation) (SA) [14]. SA and SAA are data-driven approaches, whereas the others are analytical or mixed.

Despite intensive research in the field of Chance Constrained Optimization, each subfield experiences difficulties: analytical approaches suffer from extensive conservativeness, while data-driven ones often are prohibitevely demanding in computational resources for large scale problems. The aforementioned drawbacks are critical in applications that require regular optimization. One of such applications is power systems, where a higher amount of renewable generation increases power grid uncertainty, compromises its security, and challenges classical power grid operation and planning policies [15]. It is worth mentioning that installation of renewable energy based generation is a common trend and brings substantial amount of randomness into the system [15], [16]. For example, according to recent International

Energy Agency (IEA) report [17], renewable energy generation share growth is up to 50% and to take 30% of all conventional electricity generation around the world.

In power systems applications, the aforementioned drawbacks are crucial. Con-servativeness leads to higher power market prices which influences industry and society, computational efficiency must be such that the computations accomplish within the re-evaluation interval defined by local system operator [15; 18; 19].

Relevance of the work. The research holds significant importance as it is developing and applying advanced methods such as mirror descent based adaptive importance sampling and A-priori Reduced Scenario Approximation (AR-SA), the research aims to improve the efficiency and accuracy of sample based probability estimation and optimization under uncertainty. These methods contribute to overcoming limitations in current approaches, such as impracticality for real-time operations, overestimation of risks, and computational infeasibility in large-scale problems. The outcomes of this research are expected to enhance the efficiency of commonly used approaches, making them significantly less data demanding. Moreover, the proposed methods and approaches are shown to be effective in power systems applications as they addresses critical challenges in modeling power systems with high amount of installed renewable energy generation, which become ubiquitous.

Dissertation goals. This research aims to develop advanced methods for improving the efficiency and accuracy of reliability assessments and optimization under uncertainty. The results are demonstrated on power systems applications. To achieve this goal, the following tasks were set up and performed:

1. Probability Estimation: Develop numerically efficient methods for estimating the probability measure of a polyhedron's complement. Provide theoretical support by formalizing the convergence theorem for estimate's variance and proving it. Support the theoretical advances by demonstrating algorithm's efficiency on power systems test cases. The power system example is the estimation of the probability that the current power system state is feasible against uncertainties that come from RES generation.

2. Linear Programming under Additive Uncertainty: Develop statistically based algorithms for the construction of data-efficient Scenario Approximation (SA) for Chance-Constrained Linear Programs that yield reliable

solutions, assuming additive Gaussian uncertainty for decision variables. Provide theoretical guarantees that show the improvement for required number of samples for obtaining reliable solution. Demonstrate the approximation method validity numerically on power systems test cases, compare with existing approaches. The power system example is Chance Constrained Optimal Power Flow.

3. Linear Programming under Mixed Multiplicative-Additive Uncertainty: Generalize results for a non-Gaussian source uncertainty on a non-Gaussian case, consider mixed multiplicative-additive uncertainty. Assuming typical total uncertainty mitigation setup, derive analytical condition for filtering redundant scenarios in this setting. Prove that such scenario filtering increases data efficiency and demonstrate numerically the superiority to the scenario reduction methods and ambiguous chance constraints approach. Provide demonstration on power system example which is sequential time-stamp power system modeling, where major uncertainty contributors in grid are renewable energy sources (photo-voltaic panels, wind farms, hydro electrical generators, stochastic demand) and the algorithm for uncertainty mitigation is linear Automatic Generation Control (AGC).

Scientific novelty. The scientific novelty is built up from the following results:

1. The application of adaptive importance sampling combined with mirror descent methods to power systems, particularly in scenarios involving rare events. The convergence of the algorithm was established through stating and proving a convergence theorem for optimizing estimate's variance. Next, the performance of this novel approach with practical algorithms, such as pmvnorm, was demonstrated on an example from power systems that highlights its effectiveness and potential advantages.

2. A novel method for constructing scenario approximations, introducing a more efficient and accurate approach to scenario approximation. The theoretical guarantees for this novel construction method were provided, ensuring its mathematical soundness and reliability. Lastly, a numerical demonstrations on power systems examples were conducted to compare the performance of this new scenario approximation method with classical Monte Carlo-based approach, highlighting its efficiency and accuracy.

3. Introducing an a priori approach to reduce scenario approximations for dynamic optimal power flow with automatic generation control (AGC), enhancing computational efficiency and accuracy, studying normality of generation-demand mismatch using real time series of load, renewable generation of various sources. The statements that ensured validity of the reduction approach and shown an advantage in scenario complexity were proposed. These statements confirm the method's effectiveness and reliability in reducing scenario approximations for chance constrained linear programs with multiplicative uncertainty. A numerical demonstration was conducted on a dynamic optimal power flow problem. The demonstration compares the proposed a-priori approach with other scenario reduction methods, incorporating data-driven distributional robust optimization to showcase its superior performance and practical applicability.

Theoretical and practical significance. This innovative approach could significantly advance the field of scenario approximation by offering a more efficient and accurate methods. It may lead to advancements in mathematical modeling and optimization techniques. Implementing these new methods could enhance the performance of power system simulations, leading to more accurate predictions and better decision-making in energy management. Providing theoretical guarantees for the new construction method solidifies its mathematical underpinnings, paving the way for its wider acceptance in research and practical applications. Assurance of the method's validity offers confidence to practitioners and decision-makers in utilizing it for scenario approximation in power systems, potentially leading to more reliable system planning and operation. Conducting numerical demonstrations and comparisons contributes to the theoretical understanding of scenario approximation methods, offering insights into their strengths and weaknesses. By comparing the new approach with classical Monte Carlo-based methods, the study can inform practitioners about the performance differences, aiding them in choosing the most suitable method for scenario approximation in power systems. Moreover, numerical experiments included comparison of total execution time with current state-of-the-art methods and show the practical advantages of the proposed methods.

Research methodology. Methodology included methods of linear algebra, probability theory, mathematical statistics, numerical optimization methods, aspects of optimization methods, software development and models of power systems.

Propositions for defense.

1. A numerical iterative method for estimation of a Gaussian volume of a polyhedron's complement. The method is of adaptive importance sampling family, where the sampling distribution is a Gaussian mixture. It iteratively minimizes the variance of the estimate over mixture weights using Mirror Descent. The convexity of the optimization problem is shown, stochastic gradient's expression is derived and, finally, the iterative method's convergence is shown. The method's performance is demonstrated against other estimation algorithms on power systems examples.

2. An algorithm for construction of Scenario Approximation for linear programming with additive uncertainty is proposed for solving Joint Chance Constrained programs. This algorithm is based on Importance Sampling, where the sampling distribution is Gaussian mixture. The subset of feasibility set is derived based on Gaussianity assumption for the sampling outside of it and constructing the SA based on those scenarios. The theorem on solution reliability and the number of samples required of such importance sampling based SA is stated, the proof was provided. The numerical demonstration is carried out on power systems test cases and compared to classical SA construction algorithms.

3. An algorithm for construction of Scenario Approximation for linear programming with mixed additive-multiplicative uncertainty is proposed for solving Joint Chance Constrained linear programs. The subset of feasibility set is derived for a-priori elimination of the redundant scenarios. The demonstration is conducted on the power system example which is sequential time-stamp power system modeling, where major uncertainty contributors in grid are renewable energy sources (photo-voltaic panels, wind farms, hydro electrical generators, stochastic demand) and the algorithm for uncertainty mitigation is linear Automatic Generation Control (AGC). The theorem on solution reliability and the number of samples required for reduced problems is stated, the proof was provided. The numerical demonstration is carried out on power systems test cases and compared to advanced scenario reduction methods and ambiguous chance constrained method.

Validation of the research results, reliability. The main results of the work have been reported in the following scientific conferences and workshops:

1. INFORMS Annual Meeting 2021, 1st INFORMS Workshop on Quality, Statistics & Reliability, October 15, 2021, Indianapolis

2. Rank A, CDC 2021, 60th IEEE Conference on Decision and Control, December 13-15 2021

3. Energy Research Seminar, Skoltech, October 12 2021

4. Rank A, IEEE PowerTech, Belgrade, June 25-29 2023

5. Seminar Lab. 7 of Institute for Control Sciences of Russian Academy of Sciences "Theory of Automatic Control", October 15 2024

The proposed methods and approaches were equipped with theoretical statements, the proofs were provided, numerical examples show their validity. Specifically, for iterative methods, the convergence theorem was stated. The statement introduces a bound for the target estimate's variance and reveals asymptotic behaviour with respect to the number of iterations. For those methods that are applied to Scenario Approximations, theorems on an improvement in requires number of data samples (scenarios) are stated. Finally, all the proposed methods are compared with existing methods for solving similar optimization problems using measurable metrics.

All of the proposed methodologies and approaches were published in WoS, Scopus indexed journals of rank Q1 and presented at reputable international conferences.

Acknowledgments

The dissertation was completed at the Autonomous Non-Profit Organization for Higher Education

«Skolkovo Institute of Science and Technology».

I thank my advisor, Elena Gryazina, for providing guidance and assistance at all stages of my doctoral study. I would like to devote special and sincere gratitude to my former scientific advisor and long-term co-author Yury Maximov, who initally showed the research potential in the field of statistical methods in power system and supported me throughout the whole research started even before my graduate study period. I also thank my coauthors, Deepjyoti Deka, Mile Mitrovic, Petr Vorobev, Aleksandr Bulkin and Vyacheslav Gorchakov, for numerous insightful discussions. Last but not least, I thank my wife, my family and my friends for immense support during my scientific journey.

Заключение диссертации по теме «Другие cпециальности», Лукашевич Александр Леонидович

5.7 Conclusion

Data-driven approximations are useful in chance-constrained stochastic programs with unknown uncertainty distribution and/or JCC settings. However, the data requirements rapidly become infeasible with the increase in size and reliability requirements. To address this, we proposed a novel approach that allows us to a-priorily identify and remove redundant scenarios in stochastic approximations for JCC dynamic multi-timestamp DC-OPF. We prove the validity of this approach theoretically and ensured its high empirical performance over various test cases.

Conclusion

In this thesis, we addressed the the challenges that occur in modeling power systems that experience high level of renewable energy penetration. Our main objective was to develop advanced statistical methods for current operating point reliability assessment and to propose novel optimization techniques for chance-constrained optimal power flow in various settings.

This research led to the development of adaptive importance sampling methods for grid reliability estimation and proposed new techniques for constructing scenario approximation for static and dynamic formulation of optimal power flow.

The findings have significant implications on computing and estimating generation regimes of power grids, allowing for non-restrictive robust, statistical based calculation of generation regimes and efficient estimation of the latter's reliability. This contributes to safer transition to sustainable energy solutions, contributing to global efforts to reduce greenhouse gas emissions and improve energy security. Furthermore, the proposed techniques can be applied to other areas of power systems engineering and beyond, where uncertainty arises in optimization problems, offering a foundation for future advancements in the field.

Though the methods offer substantial improvements, there are limitations to consider. The method rely on high-voltage assumption, leading to a linear system/problems. However, the results can be generalized for non-linear cases by additional mathematical effort.

Future research could explore applications for non-linear settings, broader distribution class support and wider range of engineering applications since data-driven stochastic optimization is ubiquitous at this moment. The former could be achieved by iterative constructions of convex restrictions and the latter with distributionally robust optimization techniques.

In conclusion, this thesis contributes to the ongoing efforts to integrate renewable energy sources into power systems more effectively. The developed methods not only address current challenges but also pave the way for future innovations. As the global energy landscape continues to evolve, the insights gained from this research will be instrumental in shaping a sustainable and resilient energy future.

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