Hydrothermal Scheduling in the Continuous-Time Framework

Continuous-time optimization models have successfully been used to capture the impact of ramping limitations in power systems. In this paper, the continuous-time framework is adapted to model flexible hydropower resources interacting with slow-ramping thermal generators to minimize the hydrothermal system cost of operation. To accurately represent the non-linear hydropower production function with forbidden production zones, binary variables must be used when linearizing the discharge variables and the continuity constraints on individual hydropower units must be relaxed. To demonstrate the performance of the proposed continuous-time hydrothermal model, a small-scale case study of a hydropower area connected to a thermal area through a controllable high-voltage direct current (HVDC) cable is presented. Results show how the flexibility of the hydropower can reduce the need for ramping by thermal units triggered by intermittent renewable power generation. A reduction of 34% of the structural imbalances in the system is achieved by using the continuous-time model.Comment: Accepted for publication through the Power Systems Computation Conference 202

Modelling Approaches for Hydro-Dominated System Balancing

The generation mix in many power systems across the world is rapidly shifting towards higher degrees of variable renewable power generation. Maintaining the continuous power balance of the system under higher levels of uncertainty and variability requires better planning and use of the available flexible resources. Norwegian hydropower is one such flexible resource, and the ongoing construction of substantial transmission capacity to neighboring areas with high penetration of wind and solar generation can enable the use of the hydropower as a balancing resource for the North European region. Understanding the interaction between hydropower, thermal generation, and variable renewable power sources under conditions of uncertainty is paramount in order to effectively and securely plan the operation of the system. The work conducted in this thesis has investigated the modeling of the shortterm production and reserve scheduling of systems with considerable amounts of hydropower. A fundamental approach to modeling the different power imbalances that create the need for reserve capacity and subsequent balancing energy has been adopted in all of the models. Representing large-scale cascaded hydropower in the models has been essential and has motivated the development and adaptation of several different optimization methodologies. The work has been disseminated in four scientific papers, three published and one submitted for review at the time of writing, which constitute the backbone of the thesis. The thesis includes a thorough discussion and literature review on the core topics considered in those papers. A summary of the different models created and the main results of the work performed during this PhD is provided below: • Two-stage models based on stochastic, robust, and hybrid uncertainty formulations were developed to investigate the impact of reserved capacity on hydropower plants being activated due to forecast errors in the net-load. The hybrid stochastic-robust model was found to be a good compromise between cost optimality and protection from extreme events. The distribution of reserves among the different hydropower plants is noticeably different in a deterministic model that does not consider the delivery of balancing energy, which impacts the cost of balancing the system. This effect is due to the strong temporal and topological coupling between hydropower plants created by the cascaded hydropower topology. • A hydrothermal model formulation using the continuous-time optimization framework was developed, where several modifications to the previously published continuous-time unit commitment problem were made to accommodate the inclusion of hydropower. The structural imbalances created by the discrete spot market clearing are not present in the continuous-time formulation since the power balance is kept at all times. The continuous-time model can potentially be used to estimate the cost of removing structural imbalances when compared to a standard discrete-time model. The possibility of adding continuous ramping constraints for thermal units shows how Norwegian hydropower can be used to alleviate ramping scarcity in neighboring areas. • Combining stochastic optimization and the continuous-time formulation creates a model that can capture imbalances created by both the market structure and forecast errors. The results from a stylized case study of Northern Europe with uncertain offshore wind power show that Norwegian hydropower is a principal provider of reserve capacity and balancing energy in the system. The cost increase compared to an analogous discrete-time model is roughly 0.4% of the total daily expected system cost, which stems from balancing sub-hourly wind and load variations and employing more accurate thermal ramping constraints and startup/shutdown procedures. In addition to the research published in the papers, the thesis includes Appendix C, which is a valuable resource for anyone interested in understanding and implementing the continuous-time formulation. The material in the appendix is based on the published literature on the topic and personal experience, and was written because no other comprehensive introduction to continuous-time unit commitment exists at the time of writing

Signatures of the Transition between Topologically and Magnetically Ordered Ground States in the Schwinger-Boson Mean Field Theory of Frustrated Quantum Antiferromagnets - An analysis of the triangular lattice Heisenberg model with nearest and next nearest neighbour interactions

We present three different methods of finding the critical spin value $S_c$ of the quantum phase transition between the spin liquid and the Néel order phase on the mean field triangular Heisenberg antiferromagnet at $T=0$K. These methods are: using the sublattice magnetization as an order parameter, looking at the scaling of the energy difference of topologically degenerate states in different phases, and the quantum fidelity approach. All of these methods are able to pick up the signal of the phase transition, and their estimated numerical values of $S_c$ agree for systems where the relative strength of next nearest neighbour interaction compared to the nearest neighbour interaction is small. For the relative interaction strength $j=0$ and $j=0.1$, all three methods predict $S_c\approx0.21$ and $S_c\approx0.25$ respectively, while $j=0.2$ gives deviating results. This deviation might be caused by the influence of other magnetically ordered states that we have not accounted for in our calculations, and so we restrict the validity of our results to $j<0.125$

Operating a Battery in a Hydropower-Dominated System to Balance Net Load Deviations

Decreasing costs of battery storage technologies make them a viable option for providing flexibility in power systems. In this paper, we study the operations of a battery in combination with cascaded hydropower in a system with uncertain net load due to wind power production. By simulating the operations of the combined system for a whole year by solving daily stochastic planning and real-time balancing problems, we show that the battery is only used when the hydropower system is under stress from large amounts of inflow and limited available storage capacity. The annual cost savings of adding the battery, including battery degradation costs, is 3,314 e, which is low compared to the investment costs of the battery. Since the inflow to the system follows a predictable seasonal pattern, the system operator should consider renting the battery storage from external sources, such as electric vehicles, instead of investing in a permanent battery

Fundamental Multi-Product Price Forecasting in Power Markets

This report has been prepared in the first phase of the project 'Pricing Balancing Services in the Future Nordic Power Market1, and its purpose is two-fold. First, a review on expected future trends in European power markets is provided, focusing on the sequences and rules of existing and expected future markets. Second, a literature review on the topic of fundamental price forecasting is given. The focus is on modelling approaches for short-term physical markets, including several markets, such as the dayahead intraday and balancing markets