As the power system grows more complex and active, equivalent models have become a solution for modelling parts of the network that have limited observability or are confidential or too complex to simulate otherwise. In the past decade, this topic has also made its way to distribution networks because of its transition towards an active network, which was not the case before. The current grey- and black-box techniques for equivalent modelling create models that are fitted to the average dynamic response of the system or have limited applicability. Also, the existing methods lack extensive verification under different system conditions, and these works rarely focus on active distribution networks (ADNs) with multiple points of common coupling (PCCs). This thesis aims to develop an equivalent model that can estimate the dynamic response of an active distribution network with multiple PCCs and under diverse operating conditions and topological changes.

The proposed approach is based on a graph-time convolutional neural network (GTCNN) that relates available PMU measurements inside the distribution network on a graph structure 𝒢_GTCNN. The graph 𝒢_GTCNN is obtained by taking a modified line graph of the graph representation of the power system and is expanded using the Cartesian product graph rule to include the temporal dependencies of nodes on their past values. The inputs of the equivalent model are the voltage magnitude |V| and angle θ at the PCC and the initial power injections P₀ and Q at non-PCC nodes, while the model outputs the active P and reactive Q power at the PCC nodes. The GTCNN explicitly considers the initial power injections P₀ and Q₀ at non-PCC nodes to help the model learn how different operating conditions and topological changes impact the dynamic response. The DSO trains the equivalent model using simulation data or collected PMU measurements. The model is exchanged with the TSO every month, who can use the equivalent in co-simulation with their transmission network model to perform transient stability studies.

The GTCNN-based equivalent model showed promising performance as an equivalent model for transient stability. The GTCNN was benchmarked against two state-of-the-art Long Short-Term Memory (LSTM)-based equivalent models and a hybrid GTCNN-LSTM model. The evaluation was performed on a real Dutch distribution network using three datasets, each focussing on a different system condition: different fault events, different operating conditions and different hidden topological changes. The GTCNN-based equivalent model had a mean-squared error (MSE) below 0.02 for each dataset, which means it can accurately reproduce the dynamics. This accuracy is comparable to the LSTM-based equivalent models, but the GTCNN could train 4x faster. The GTCNN also showed good generalisation performance, as its accuracy did not decrease on the validation and test sets. A study on scaling performance suggested that the MSE of the GTCNN-based equivalent model increases slower than that of the LSTM-based models while its training time increases faster. Therefore, the GTCNN-based equivalent model trains faster for smaller ADNs but will be more accurate with more measurement nodes. However, the proposed GTCNN has difficulty learning the response at different close-by PCC terminals if the dynamics are different.

The developed GTCNN-based equivalent model can predict the dynamic response accurately under changing topologies and operating conditions at a similar performance level to existing LSTM-based approaches. However, its training time is much faster, which can result in a more accurate equivalent model by a more frequent model exchange between the DSO and TSO or a more extensive dataset being used to train the model. In future research, the GTCNN performance will be evaluated on a more comprehensive dataset containing all three system conditions to establish how much data is needed to train the equivalent model accurately. Also, the system frequency will be considered as an additional input. Moreover, its scaling performance will be evaluated more extensively and with a more efficient coding implementation. Furthermore, a heterogenous graph convolutional operator will be implemented to learn the connection per relational type (source node - edge type - target node). Finally, the co-simulation interface between the equivalent model and popular simulation tools will be explored.

To keep pace with increasing renewable energy penetration and consequent increase in inverter-based resources in the power grid, it is pertinent for present-day research to address the resulting drop in system inertia levels and its impact on frequency stability. With decreasing levels of inherent rotational inertia present in the system, any sudden disturbance causing an energy imbalance in the grid could lead to more drastic excursions of system frequency than those experienced hitherto. To ensure the resilience of the grid in such scenarios, advanced and competent frequency stability assessment and control methods are required. This thesis presents Neural Ordinary Differential Equations (NODE), a recently introduced family of neural networks, as an effective tool to achieve fast, real time estimates of the expected frequency response trajectory during an energy imbalance event.

Since high-impact frequency instability events are sparse in reality, both real-world grid data and synthetically generated data corresponding to different inertial conditions are used to train predictive NODE models. Firstly, NODE is adapted to frequency prediction applications through relevant data processing steps, and modification of network parameters and algorithmic aspects pertaining to the predictive model definition. Secondly, patterns corresponding to specific sections of the frequency response curve are used to selectively train NODE models. Pattern-specific training methods exhibit better prediction performance when the NODE model encounters frequency behaviour similar to the one it initially trained on. Thirdly, a pre-training approach to cut short on the real-time training time required by NODE models to achieve desired levels of prediction performance is presented. Fast estimates of critical frequency stability parameters like nadir could act as potential triggers for early stability control actions to achieve a more controlled frequency response.

Application of predictive NODE models for different frequency scenarios are presented using three test-cases: normal operating scenario, restoration post-system split scenario and synthetically generated high-impact frequency disturbance scenarios. Model tuning and training methods specific to each test-case are described, and prediction results are evaluated with relevant performance metrics. Finally, a comparison is made between the implementation of NODE among different test-cases and real-world implications of the frequency prediction outcomes from the test-cases are further discussed.

This work seeks to resolve an outstanding problem in the use of reinforcement-learning methods for the simulation of economically-rational agents. We discuss the problem of non-stationarity, and how this subsequently limits market simulation capabilities. After explicating and isolating the source of the problem for a day-ahead electricity market, we demonstrate the application of methods which resolve this problem in simple test-cases, and prove conditions under which similar methods will work in general. Subsequently, we illustrate how these techniques can be used to solve a restricted market-design problem, in the process introducing a framework for discussing adversarial market-design for electricity markets in general. It is hoped that, insofar as they provide a new feedback-loop for market-design, these results will facilitate the design of more complex electricity-markets suitable for the energy transition.

In the electricity system, one barrier to the energy transition is the degradation of frequency stability due to the decrease of system inertia and frequency control ancillary services (FCAS), which is caused by the replacement of inertia-abundant and governor-based conventional power plants with zero-inertia and inverter-based renewable energy sources (RES).

There are already inverter technologies for RES and battery energy storage systems (BESS) to provide virtual inertia (VI) and fast frequency response (FFR) services, which are equivalent to physical inertia and conventional FCAS. Potential providers include wind, solar, battery energy storage systems (BESS), and other types of devices with the feature of energy storage. However, there are non-technical barriers to the actual implementation. For example, in most parts of the world, these services cannot participate in the electricity market and thus there is a lack of incentive for both the provision of and investment in the VI and FFR.

In the literature, there are already proposals of possible market designs for VI and FFR that procure the services, guarantee the frequency-stability requirements, and provide payments to the service provider. The main focal point is on the formulation of security-constrained unit commitment (SCUC) and security-constrained economic dispatch (SCED) problems. The formulation needs to be accurate in modeling, be solvable, and be with reasonable computational burden. Most of them only considered the allocation of ancillary services but did not price them, or price them but only by directly assigning a shadow price. Only a few works considered explicit prices in the bid.

In this project, we consider a market design with explicit bid prices. In the SCUC and SCED problems, we adopt the state-of-the-art formulation of frequency nadir constraint and method of modeling the frequency dynamics based on the linear ramp assumption of the dynamics of frequency response (FR). With these methodologies, we will investigate the features of such a market by identifying inter-dependencies of parameters in the market setups, including the interaction between the bid price and bid amount of FR and VI, and analyzing the underlying mechanisms. The results show that the amount of FR sold depends on the bid price of FR monotonously and the amount of VI sold depends on the bid price of VI monotonously, though with different patterns. The underlying reason for such dependencies is that FR, VI, and the size of the largest unit both help mitigate frequency drop and recover it and they are influencing each other. The price of FR or VI decides the relative worthiness of each option. The size of the largest unit is also related to the amount of FR sold, which is confirmed by the formulation of the analytical nadir and the QSS constraint. However, the way that the amount of available FR influences the amount of VI sold does not show a clear pattern.

These findings provide insights into the interactions between the FR and VI products and thus provide a reference for the design of the FCAS market.

The transition to green energy is reshaping the energy landscape, marked by increased integration of renewable energy sources, distributed resources, and the electrification of other energy sectors. These changes challenge grid security, particularly regarding the N-1 security criterion, a crucial factor in preventing blackouts. Furthermore, climate change is contributing to the growing frequency of extreme weather events, which constitute the second major cause of blackouts. As grid complexity keeps on increasing, the need for N-k security, where k lines fail simultaneously, and increased resilience against extreme weather events is becoming increasingly evident. This necessitates studying the security constrained optimal power flow (SCOPF) problem considering multiple line outages (N-k). Current methods exhibit poor scalability as k increases. In response to the challenge of limited scalability, this thesis proposes a constraint-driven machine learning approach to approximate N-k SCOPFs.

The proposed approach relies on the linearized direct current optimal power flow. The approach utilizes a neural network to map power system loads to generator setpoints. A feasibility restoration layer is employed to restore base case infeasible predictions. By incorporating line outage distribution factors (LODFs), all post-contingency flows are computed. The loss function utilized to train the neural network draws inspiration from the penalty function method. Lastly, a copula analysis computes joint outage probabilities for k \textgreater 1 enabling a probabilistic security assessment. The first academic contribution of this thesis is the development of a constraint-driven approach to approximate N-k SCOPFs considering all contingencies using LODFs. The second academic contribution is the formulation of a N-k risk based security criterion, providing an alternative to the current deterministic N-1 security criterion.

The approach shows promise in its ability to scale effectively to N-k contingencies. Using LODFs, the approach effectively computes all post-contingency flows for up to k = 3. Moreover, case studies show the constraint-driven approach's effectiveness in identifying violating post-contingency cases, with up to 173$\times$ speedups and close to optimal dispatch costs. However, the consideration of N-k contingencies holds combinatorial complexity, and more efficient methods need to be developed for the computation and storage of all LODFs, and for the computation of all post-contingency flows. Additionally, the proposed constraint-driven approach can not enforce any post-contingency constraints, necessitating post-contingency feasibility checks when security against specific contingencies is required. Next, by incorporating probabilities, the approach shows promise in improving power systems security and resilience, but further research is necessary.

In this thesis, only line outages are considered. In the future, the approach could be modified to additionally account for other equipment outages (e.g. generator outages). Furthermore, future research could investigate the adoption of this approach in corrective control settings, where it is employed in the restorative phase of a contingency event. Another suggestion is centered around the incorporation of graph neural networks in the proposed approach, which could provide a more scalable alternative to fully connected linear neural networks. Furthermore, more scalable methodologies could be explored to construct the matrix containing all LODFs, and a more scalable methodology for computing all post-contingency flows could be developed. Finally, future work could investigate how to utilize the proposed approach under varying conditions like network topology changes or changing outage probabilities.

With the increasing environmental concerns, the world is moving towards rapid decarbonization, and to meet the growing energy demand, more renewable energy sources like solar, Wind are getting added to the existing Distribution grid. The addition of new loads like Electric vehicles (EV), Heat pumps, etc. is putting additional pressure on the grid leading to some technical problems. Due to this, it is important to have real-time monitoring of the system by the Distribution System Operators(DSO)s which can be obtained by performing the State Estimation. The Distribution System State Estimation along with the state estimator has been discussed in the literature.

However, the accuracy of the state estimation is highly affected by the absence of real measurements from the meter and the presence of high-variance pseudo measurements. But due to the cost factor, it is practically not possible to install meters at every node. To reduce the state estimation error, a new meter placement strategy has been proposed in this project with the help of open-source software called Power Grid Model, devolved by the Dutch DSO Alliander for performing steady state analysis of the network.

The work presented in this project includes a sensitivity analysis of the proposed meter placement algorithm which is applied to a test MV distribution network in the Netherlands for the different sample sizes of the load profile as well as the standard deviation of the pseudo measurements. The application of the meter placement algorithm results in a subsequent reduction of the state estimation error. To find a suitable number of meters that can be placed in the network, three-meter placement criteria have been proposed and applied to the test network. The results show the effectiveness of the proposed meter placement algorithm for the determination of a suitable path as well as the number of meters that can be placed in the distribution network. This is important for the grid operators as it will help them to make decisions for the proper maintenance and operation of the grid.

The growth of renewable energy technologies is leading to energy systems that are more reliant than ever on renewables such as Wind and Photovoltaic (PV) power. Despite their benefits in terms of sustainability, their ubiquity poses challenges in maintaining grid stability given their intermittency, emphasising the prediction of power fluctuations. Physical models and statistical approaches, especially for nowcasting (forecasting for 0-6 hours in the future), have been superseded by Machine Learning (ML) methods in terms of forecast accuracy (below 3% Root Mean Squared Error (RMSE)). Within ML, Artificial Neural Network (ANN) methods seem to perform particularly well for nowcasting. This project focuses on predicting solar and wind meteorology with that level of accuracy, and on how to best use the prediction to minimize the cost of maintaining a balanced energy system, i.e. one where power consumption matches production at any moment. Producing accurate power predictions based on Multi-Modal (MM) data and the extent to which prediction accuracy reduces system cost are challenges to be addressed in this thesis. MM and End-to-End (E2E) training (with the system cost as the task of an ANN based algorithm) are investigated to this end. MM learning involves handling information from multiple types of input (audio and visual, for example) for performing a ML task such as regression or classification. It is of interest for this project because it has been shown to outperform other NN approaches in predicting sudden changes in solar irradiance. E2E learning entails an algorithm design which predicts the end goal of a ML process directly from the inputs. This is pursued because it addresses the true task (cost minimization) of system operators as the focus of the ML algorithm. The proposed method consists of a NN architecture that learns to fuse features from MM data (sky imagery and meteorological sensor data) at intermediate layers of the network in order to predict PV or Wind generation. This prediction is then used as an input to an Optimal Power Flow (OPF) problem (which seeks to minimize generation costs in a power system, considering power balance and transmission network constraints to ensure the twin goals of economic and secure system operation). The proposed model is trained E2E, therefore it is informed by the minimized cost solved by the optimization, rather than the intermediate power prediction (as conventional approaches would involve). In an IEEE 6-bus system with PV generation, a sequential training baseline results in costs 10% higher than a perfect forecast, while our proposed MM4-E2E approach achieves costs only 7% higher, a significant improvement. The intermediate prediction of PV power by MM4-E2E is also improved, with 18% lower RMSE by the proposed model compared to the baseline, explained by the enhancement of one modality by the other through MM learning. In a power system with two renewable sources, costs are reduced through the proposed model compared to a conventional approach (4% excess cost compared to 7%, measured against a perfect forecast), but power prediction accuracy is worse, sue to convergence to local minima.

Renewable energy sources, although they are quickly increasing their share in the energy mix, face a major barrier to more widespread adoption. Energy storage solutions overcome this hurdle, and lithium-ion batteries are at the forefront of this. The need for lithium-ion battery degradation studies arises due to the ever increasing use of these batteries in a wide variety of applications.

There exist a large number of empirical and semi-empirical battery degradation models in literature, however their usage with irregular, real world profiles has not been explored. Moreover, understanding where different models can be used and comparing these models is also an area that has not been looked at.

A real world power profile was created in MATLAB based on the WLTP drive cycle. Simultaneously, each model considered was verified against the experimental data for that same model to make sure model predicted degradation values matched the experimental observations. In three models, the verification procedure revealed errors and inaccuracies that were corrected. Implementing an irregular
power profile posed two challenges, first, the stress factors of time and throughput required linearized versions of the model equations to be properly accounted for. Second, the identification of cycles in the irregular power profile is usually done using the rainflow counting algorithm, however, an alternative method is proposed in this study.

The real world power profile was applied to each calendar and cyclic ageing model and it was noted that although the power profile was equalized for each cell based on the model in question, the results were widely differing. In some cases, it was possible to explain the differences but in some cases the differences were not easily explained. Apart from the comparison, each model was individually analyzed to understand how degradation phenomena and stress factors affect the accuracy and use case of a model.

Finally, a toolbox was built in MATLAB to summarize the findings of this study, to help users understand where each model studied is likely to be accurate and useful. It was concluded that empirical and semi-empirical models are highly dependent on the testing conditions of the experimental data they
are built on, and there are extremely limited scenarios in which these models are applicable outside these conditions. Moreover, accelerated testing conditions which are often used in the experimental phase usually cover different degradation phenomena than those which occur under regular use cases.
With respect to the application of irregular (real-world) power profiles to these models, this study details a unique method used to obtain accurate predictions.

Short-term solar forecasting is crucial for large scale implementation of solar energy and plays an important role in grid balancing, energy trading, and power plant operation. Cloud movement is the main source of unpredictability within solar forecasting and can be recorded using All-Sky Imagers. Conventional cloud modelling methods using image analysis techniques are unable to extract the spatial configuration and the temporal dynamics of clouds, resulting in poor predictions of the interaction with solar radiation. The goal of this study is to create a deep learning model for short-term irradiance forecasting between 0 and 21 minutes into the future using all sky images combined with auxiliary data. The model performance was assessed by comparing the deep learning model with the persistence model and showed that the deep learning model outperforms the persistence model with 24.8%. A sensitivity analysis to data usage is performed showing that besides using more data, also the variation of using multiple years of data results in better performance. Furthermore, the sensitivity of the model to input variables is assessed, showing that using the clear sky irradiance as input improves model performance with 16% and that meteorological data does not improve performance. Additionally, the model performance was evaluated during different sky conditions showing that the deep learning model outperforms the persistence model for all sky conditions, except overcast conditions. An example of the model behavior is extensively described, showing that the deep learning model tends to predict the trend of the irradiance fluctuations rather than the actual fluctuations. Next to that is in this study shown that the current deep learning model occasional miss important weather events, like obscuration of the Sun, resulting in large irradiance prediction errors. A pathway for future improvements for deep learning models to forecast the short-term irradiance is provided.