With the increasing integration of power plants into the frequency-regulation markets, the importance of optimal trading has grown substantially. This paper conducts an in-depth analysis of their optimal trading behavior in sequential day-ahead, intraday, and frequency-regulation markets. We introduce a probabilistic multi-product optimization model, derived through a series of transformation techniques. Additionally, we present two reformulations that re-frame the problem as a mixed-integer linear programming problem with uncertain parameters. Various aspects of the model are thoroughly examined to observe the optimal multi-product trading behavior of hydro power plant assets, along with numerous case studies. Leveraging historical data from Nordic electricity markets, we construct realistic scenarios for the uncertain parameters. Furthermore, we then proposed an algorithm based on the No-U-Turn sampler to provide probability distribution functions of cleared prices in frequency-regulation and day-ahead markets. These distribution functions offer valuable statistical insights into temporal price risks for informed multi-product optimal-trading decisions.

For inclusion in large-scale power system models, various aggregations and simplifications in the modeling of relevant actors are needed. This paper focuses on reduced models of hydropower, so called area Equivalent models. They use a simplified topology but are not a direct aggregation of the real hydropower system. Instead, the area Equivalent is constructed to mimic the simulated power production of a more detailed hydropower reference model. Here, this goal is fulfilled by formulating a bilevel problem minimizing the difference in simulated power production between the area Equivalent and its reference. Solving this can be computationally heavy. Thus, for a fast solution of this bilevel problem, a single-level reduction is done, which is then solved using two methods. The first method includes McCormick envelopes to form a linear single-level problem. Second is a modified Benders with a relaxed sub-problem to handle the non-convex single-level. These are then also compared to Particle Swarm Optimization. Moreover, six new upper-level objective functions are investigated for a case study of hydropower in northern Sweden. The method using McCormick envelopes is fast (2–5 min), but the area Equivalent shows lower average performance. The modified Benders finds a solution in 5–31 min with good performance.

To simulate a hydropower system, one can use what s known as a Detailed model. However, due to the complexity of river systems, this is often a computationally heavy task. Equivalent models, which aim to reproduce the result of a Detailed model, are used to significantly reduce the computation time for large-scale hydropower simulations. This paper computes Equivalent models for hydropower systems in Sweden by categorizing the water inflow data using a spectral clustering method. Computing the Equivalent models is done using a variant of the particle swarm optimization algorithm. Then, the Equivalent models are evaluated based on their similarity to the Detailed model in terms of power production and objective value. The Equivalent models range from 8% - 12% error in terms of the relative power production difference and the computation time is reduced by at least 99.9%.

With over 4000 TWh yearly electricity production worldwide, hydropower plays an important role in many power systems. Unlike many other renewable energy sources, hydropower has a certain degree of controllability and high levels of flexibility over several time scales. This flexibility is estimated to be integral for the transition of the energy systems towards more variable renewable energies and thus reducing greenhouse gas emissions. 

Given the important role that hydropower currently plays and is expected to play in future power systems, accurate models of hydropower are vital. As hydropower electricity production is a non-convex function of the discharge with for example non-linear head dependencies and forbidden zones of operation, detailed models of real hydropower systems quickly become computationally heavy. Even linear models with high numbers of interconnected stations are often too complex for large-scale power system models. For this reason, reduced or aggregated models of hydropower are commonly used to simulate its operation in different power system models. 

Due to the temporal and spatial connections in many hydropower systems with large rivers, the aggregation of hydropower can pose significant challenges. This means that aggregation from historical data might not be good enough to accurately simulate the hydropower operation. However, accurate reduced models of hydropower are still needed for long-term current and future studies of energy systems worldwide. In this thesis, the basic assumption is that the simplified reduced hydropower model should mimic the real hydropower operation. Thus, instead of aggregating the existing hydropower stations within a certain geographical area, one computes a new hydropower area Equivalent model with the aim to match the simulated power production of a more Detailed model of the real hydro system in that area. 

In this work, the area Equivalent models are calculated by computing the model parameter values. Here, this is mainly done based on a bilevel optimization problem formulation. In this thesis, different methods to compute the area Equivalents are proposed together with different model formulations and bilevel problem formulations. These are all compared using case studies of Swedish hydropower systems. Moreover, a Baseline aggregation method is outlined and compared to the developed area Equivalents. 

The studies presented in this thesis highlight the potential trade-offs in the accuracy of the area Equivalent model. Some problem formulations give a higher accuracy in hourly power production, others in peak power production or total power production over the simulation period. All area Equivalents perform better than the Baseline aggregation. In general, the average error in hourly power production is reduced by 50% using the area Equivalent compared to the Baseline aggregation. Moreover, they all successfully reduce the simulation time compared to the reference Detailed model with over 96%.

Med mer än 4000 TWh årlig elproduktion värden över, spelar vattenkraft en viktig roll i många kraftsystem. Till skillnad från många andra förnyelsebara energikällor har vattenkraft en viss grad av styrbarhet och en hög nivå av flexibilitet över flera tidsskalor. Denna flexibilitet kan antas ha en stor betydelse för energisystemens övergång till mer varierande förnyelsebar energi och därmed även minska växthusgasutsläppen. 

Givet den viktiga roll som vattenkraften har idag och även väntas fortsätta ha i framtida kraftsystem så är exakta modeller av vattenkraft nödvändiga. Eftersom vattenkraftens elproduktion är en icke-linjär funktion av tappningen med till exempel icke-linjära fallhöjdsberoenden och förbjuda driftszoner, blir detaljerade vattenkraftsmodeller snabbt beräkningstunga. Även linjära modeller med många sammankopplade stationer är ofta för komplexa för storskaliga kraftsystemsmodeller. Till följd av detta används ofta reducerade eller aggregerade modeller av vattenkraften för att simulera dess drift i olika kraftsystemsmodeller. 

På grund av tidsmässiga och hydrologiska kopplingar i många vattenkraftssystem med stora älvar kan aggregering av vattenkraft utgöra signifikanta utmaningar. Det innebär att en aggregering baserat på historiska data kanske inte är tillräckligt bra för att exakt simulera vattenkraftsdriften. Likväl behövs fortfarande exakta reducerade modeller av vattenkraft för långsiktiga studier av nuvarande och framtida energisystem världen över. I den här avhandlingen är det grundläggande antagandet att den förenklade och reducerade vattenkraftsmodellen ska spegla den verkliga vattenkraftsdriften. Därför, istället för att aggregera de existerande vattenkraftsstationerna inom ett visst geografiskt område, ska man beräkna en ny vattenkrafts-area-Ekvivalent som har som mål att matcha den simulerade kraftproduktionen från en mer Detaljerad modell av det verkliga vattenkraftssystemet i det området. 

I det här arbetet beräknas area-Ekvivalenterna genom värdena på modellparametrarna. Här görs detta främst genom ett optimeringsproblem med två nivåer. I den här avhandlingen föreslås olika metoder för att beräkna area-Ekvivalenterna tillsammans med olika modellformuleringar och formuleringar av två-nivå-optimeringen. Alla dessa jämförs i olika fallstudier av svenska vattenkraftssystem. Dessutom beskrivs en metod för en Bas-aggregering som också jämförs med de utvecklade area-Ekvivalenterna. 

Studierna presenterade i den här avhandlingen visar på potentiella avvägningningar mellan olika typer av exakthet hos den Ekvivalenta modellen. Vissa problemformuleringar ger en högre grad av exakthet i timproduktion, andra i topproduktion eller total produktion över simuleringsperioden. Alla områdes Ekvivalenter presterar bättre än Bas-aggregeringen, överlag minskar felet i timproduktion med 50% för områdes Ekvivalenterna jämfört med Bas-aggregeringen. Dessutom lyckas de alla minska simuleringstiden jämfört med den Detaljerade referensmodellen med över 96%.

Modeling large energy systems requires different forms of simplifications and aggregations. This is especially true for large hydropower systems. One way to simplify the modeling of hydropower as a part of large scale energy systems is to utilize so-called Equivalent models. The hydropower Equivalent model is a simplified hydropower area model with only one (or a few stations) which aims to mimic the behavior of an Original more detailed model containing all stations in a specific area. However, one drawback has been that the Equivalent model fails to match the highest production peaks of the real Original system. Methods to increase the maximum peaks in the Equivalent model have so far resulted in overall lower performance, where the production during lower peaks instead would be overestimated. Thus, in this paper, a method for computing hydrosystem area Equivalent models that not only have good average performance but also can capture the production peaks of the real hydropower system is developed. The new method allows for optimal partition and efficiency of different segments in the hydropower marginal production function.

Simulation of sizeable hydro-thermal power systems, such as Northern Europe or larger, requires several extensive simplifications and model reductions to decrease simulation time. Such reductions for hydrosystem sare often called Equivalent models. Their purpose is to mimic a more detailed hydropower model whiled ecreasing computation time. Both aspects are vital for accurate and useable simulation results. Here, different Equivalent models for hydropower have been developed together with a new function for adaptive Equivalentinflow based on local inflows to the detailed system. The models were computed via a bilevel optimization problem factoring in the novel adaptive inflow. Based on this, the new function for adaptive inflow was calculated using regression. The Equivalents have then been evaluated in a case study of hydropower systems in Northern Sweden regarding accuracy in hourly and total power generation, revenue estimation, and relative computation time. For all Equivalents, the computation time is decreased by >96%. Further, the Equivalents demonstrate improved performances in hourly and total power production and revenue estimations. The best hourly power difference was 9.2%, and the best revenue estimation was 5.9%. Especially notable is the low total power production difference of <0.5% compared to the more detailed model.

Simplified models of hydropower systems are necessary for simulation of large power systems, long-term analysis, and future studies. One common simplification has been to aggregate all hydropower within an area based on historical data. Another option is to use mathematical so-caned hydropower Equivalents. Here, hydropower Equivalents represent an optimized model reduction of a more Detailed model depicting the complete hydropower system within a specific area. These Equivalents are computed based on a bilevel optimization problem formulation. In this paper, the impact different Equivalent model constraints have on the performance is analyzed via a novel investigation of new model formulations. Moreover, recent solution methods and a baseline aggregation of the hydropower from statistics are compared and evaluated for the first time. All bilevel Equivalents show a significantly better performance than the baseline aggregation; the accuracy in hourly power generation relative to the Detailed model is almost twice as high for all bilevel Eqmvalents.

Actual hydropower systems often contain complex river systems, making simulations of hydropower systems computationally extensive. With the help of an Equivalent model, which is an artificial simpler hydropower system, the computational issue can be avoided. The Equivalent model aims to imitate the actual hydropower system, hence can be used for studies of large power systems with faster computational time. In this paper, different Equivalent models of hydropower systems in different periods and trading areas in Sweden have been computed, which are then evaluated and validated by comparing the hourly relative power generation difference. The relative power difference is 4.89% - 19.36%, depending on the period and trading area. Different Equivalent model performance over time indicates that the Equivalent parameters should change over time. By dividing a year appropriately, a well performing Equivalent can be obtained, which is shown by the Equivalent with low relative power difference.

Hydropower system Equivalents are fictive model reductions of a more detailed hydropower model of all individual hydropower stations within an area. The system Equivalent only has one or a few stations and can significantly reduce the computational effort compared to the detailed system. The aim is to at the same time maintain accuracy in power production. To achieve this a bilevel problem formulation can be utilized. In this bilevel problem, the optimal system Equivalent parameter values are calculated. However, this computation can be very time-consuming and the problem formulation is very complex. Moreover, which of the parameters are most important for overall Equivalent performance and accuracy is unknown. To get a better understanding of the system Equivalent model parameters and their connection to performance, this paper performs a sensitivity analysis and develops a method to determine the parameter importance. Based on these results methods to reduce the computational time of the bilevel problem can be devised, with an example given here. It is shown that the most important parameter is the maximum discharge limit.

Simulation of large hydro-thermal power systems requires several extensive simplifications and model reductions. For hydropower systems with several interconnected power stations, these reductions can be particularly challenging and are denoted Equivalent models. The purpose of the Equivalent model is to mimic a more detailed hydropower model, while decreasing computation time, to be used in larger power system models. In this paper different Equivalent models for hydropower systems have been computed with a novel approach using a Particle swarm optimization-algorithm and are evaluated with respect to accuracy in hourly and total power generation as well as computation time. For each of the Equivalent models, computation time is decreased with over 99.99 % and the difference in power production is less than 11% compared to a more detailed model.