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# PhD defense by Dídac Rodríguez Arbonès

## Title

Evolution strategies with Optimal Covariance Matrix Update Applied to Sustainable Wave Energy

## Abstract

Modern society depends heavily on fossil fuels. We rely on this source of energy for everything, from food and clothing production to daily transportation. Even the Internet is mostly powered by these sources of energy. This reliance has led us to a high-risk situation where all that we take for granted is in jeopardy. Some of the causes of modern armed conflicts can be traced back to the scarcity and struggle over these types of energy sources. Extracting energy from renewable sources can decrease our dependency on fossil fuels. There exist many types of renewable energy sources, and wave energy is a rather unexploited one. Even though humans have thought about harvesting the power of the waves for centuries, the efficient and large scale extraction of this type of energy is cumbersome. The modelling and simulation of extraction infrastructure is a challenging research task, and building sizeable plants requires large financial investments.

A common type of wave energy plants are buoy farms. These farms consist of a group of buoys moored to the sea floor. The buoys capture the movement of the waves and pump hydraulic fluid onshore, where a turbine generates power. Constructive and destructive interference between buoys makes the prediction of power output a difficult problem. By a careful positioning these interactions can be exploited to achieve higher power outputs. The choice of farm layout can be formulated as an buoy positioning optimization problem. However, due to the complex nature of wave power modelling, gradient information is not available and one has to resort to derivative-free optimization methods. Furthermore, not all possible buoy configurations can be implemented. The daily operation of wave energy plants introduces constraints on the layouts that can be achieved in practice. Moreover, objectives other than the amount of power produced often need to be taken into account. Factors like the amount of area utilized or the pipe distance to the onshore turbine are also important. In such cases, multi-objective optimization algorithms are required to obtain solutions that consider the trade-offs between these different goals. Evolutionary algorithms are optimization methods that can meet all the requirements posed by the buoy positioning problem. They have been successfully applied for constrained single- and multi-objective derivative-free optimization in many real-world applications, such as wind turbine blade geometry. In particular, the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) is widely considered as one of the best derivative-free methods for real-valued optimization.

This thesis considers methodological improvement of the CMA-ES and its application to the problem of buoy positioning for energy production. The CMA-ES is a Monte Carlo method, sampling candidate solutions from a multi-variate Gaussian distribution that is continuously updated. Given the objective function values at the sampled points, updating and storing the covariance matrix of the Gaussian distribution dominates the time and space complexity in each iteration of the algorithm. This complexity is reduced by a numerically stable quadratic-time update scheme with minimal memory requirements. The Cholesky decomposition of the covariance matrix is updated instead of the matrix itself. The proposed modification slightly affects cumulative step-size adaptation, a mechanism often used for adapting the global step-size parameter in the CMA-ES. While this has a negligible effect on the performance, the time benefit from the Cholesky update and the memory savings from storing a Cholesky factor instead of a full matrix are substantial.

Then a multi-objective variant of the CMA-ES is applied to buoy farm layout optimization and compared to other evolutionary algorithms. As the energy production model used is exponential in the number of buoys, an approximation scheme is devised to allow for experimentation with larger farm sizes. Farm size and buoy distance constraints are considered for practical purposes. These constraints further increase the complexity of the optimization problem. The results show an increase of approximately 1% for a farm of 36 buoys with respect to a naïve grid layout, which represents an expected gain of approximately 54 KW, equivalent to dozens of average households.

**Assessment Committee**

- Chairman: Associate Professor Kenny Erleben, Department of Computer Science, University of Copenhagen
- Professor Heike Trautmann, Department Information Systems, University of Münster, Germany
- Professor Oliver Kramer, Department Computer Science, University of Oldenburg, Germany

**Academic supervisor and Co-Supervisor**

- Professor Christian Igel, Department of Computer Science, University of Copenhagen
- Kurt Sig Kristensen (NETS A/S, Lautrupbjerg 10, Ballerup)

*For an electronic copy of the thesis, please contact phdadmin@di.ku.dk*