Mathematical Programming for Power Systems Operation. Alejandro Garcés Ruiz

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Acknowledgment

      Throughout the writing of this book, I have received a great deal of support and assistance from many people. I would first like to thank my friends Lucas Paul Perez at Welltec, Adrian Correa at Universidad Javeriana in Bogotá-Colombia, Ricardo Andres Bolaños at XM (the transmission system operator in Colombia), Raymundo Torres at Sintef-Norway, and Juan Carlos Bedoya at the Pacific Northwest National Laboratory (USA), who, in 2020 (during the COVID-19 pandemic), agreed to discuss some practical aspects associated to power system operation problems. The discussions during these video conferences were invaluable to improve the content of the book. I am also very grateful to my students, who are the primary motivation for writing this book. Special thanks to my former Ph.D. students, Danilo Montoya and Walter Julian Gil. Finally, I want to thank the Department of Electric Power Engineering at the Universidad Tecnológica de Pereira in Colombia and the Von Humbolt Foundation in Germany for the financial support required to continue my research about the operation and control of power systems.

       Alejandro Garcés

Introduction

      Electrification is the most outstanding engineering achievement in the 20th century, a well-deserved award if we consider the high complexity of generation, transmission, and distribution systems. An electric power system includes hundreds or even thousands of generation units, transformers, and transmission lines, located throughout an entire country and operated continuously 24 hours per day. Running such a complex system is a great challenge that requires using advanced mathematical techniques.

      All industrial systems seek to increase their competitiveness by improving their efficiency. Electric power systems are not the exception. We can improve efficiency by introducing new technologies but also by implementing mathematical optimization models into daily operation. In every mathematical programming model, we require to perform four critical stages depicted in Figure . The first stage is an informed review of reality, identifying opportunities for improvement. This stage may include conversations with experts in order to establish the available data and the variables that are subject to be optimized. The second stage is the formulation of an optimization model as given below:

      

(0.1)

Where x is the vector of decision variables, f is the objective function and, Ω is the set of feasible solutions. Going from stage one (reality) to stage two (model) is more of an art than a science. One problem may have different models and different degrees of complexity. Practice and experience are required to master this stage, as some models are easier to solve than others. Subsequently, the third stage consists of the implementation of the mathematical model into a software. After that, the fourth stage is the analysis of results in the context of the real problem.

      Figure 0.1 Stages of solving an optimization problem.

      This book will focus on stages two and three, associated with power system operations models. In particular, we are interested in models with a geometric characteristic called convexity, that present several advantages, namely:

       We can guarantee the global optimum and unique solution under well-defined conditions. This aspect is interesting from both theoretical and practical points of view. In general, a global optimum advisable in real operation problems.

       There are efficient algorithms for solving convex problems. In addition, we can guarantee convergence of these algorithms. This is a critical aspect for operation problems where the algorithm requires to be solved in real-time.

       There are commercial and open-source packages for solving convex optimization models. In particular, we are going to use CvxPy, a free Python-embedded modeling language for convex problems.

       Many power system operations problems are already convex; for example, the economic and environmental dispatches, the hydrothermal coordination, and the load estimation problem. Besides, it is possible to find efficient convex approximations to non-convex problems such as the optimal power flow.

In summary, convex problems have both theoretical and practical advantages for power systems operation. This book studies both aspects. The book is oriented to bachelor and graduated students of power systems engineering. Concepts related to power systems analysis such as per-unit representation, the nodal admittance matrix, and the power flow problem are taken for granted. A previous course of linear programming is desirable but not mandatory. We do not pretend to encompass all the theory behind convex optimization; instead, we try to present particular aspects of convex optimization which are useful in power systems operation. The book is divided into two parts: In the first part, the main concepts of convex optimization are presented, including a distinct chapter about conic optimization. After that, selected applications for power systems operation are presented. Most of the solvers for convex optimization allow mixed-integer convex problems. Therefore, we include models that can be solved in this framework too. The student is recommended to do numerical experiments in order to acquire practical intuition of the problems.

      All applications are presented in Python, which is a language that is becoming more important in power systems applications. Students are not expected to have previous knowledge in Python, although basic concepts about programming (in any language) are helpful. Our methodology is based on many examples and toy-models. We made a great effort in showing the most simple model with a clean code. Of course, these toy-models are an oversimplification of the real problem; however, they allow us to understand the model and its coding. In practice, we may have complex models that combine different aspects such as the economic dispatch, the unit commitment and/or the optimal power flow. A real operation model may require a sophisticated platform that integrates the model with the supervisory control and data acquisition system (SCADA) operating in real-time. The development of such a real industrial model is beyond the objectives of this book.

1 Power systems operation

       Learning outcomes

      By the end of this chapter, the student will be able to:

       Identify problems related to power systems operation.

       Link mathematical optimization models to power systems operation problems.

      1.1 Mathematical programming for power systems operation

      Mathematical optimization is a fundamental tool for the electrical supply chain, from generation through transmission, distribution, and end-use. It may also be used, in different time frames, from a few milliseconds to several years. This book concentrates on optimization problems for power systems operation. These problems are usually continuous and have a time frame from several minutes to one day. Optimization problems with faster dynamics lie in the control and stability analysis, whereas problems with slower dynamics are planning problems.

Mathematical