This paper presents a dynamic phasor-based state-space modelling framework for modular multilevel converters (MMCs) using multiple dq reference frames (DQsym). The main contribution is the development of a numerically stable state-space solver tailored to multi-dq dynamic phasor models. The solver employs integration and a systematic matrix update mechanism to consistently incorporate switching events within the unified state-space formulation, allowing harmonic interactions and topology changes to be captured without manual reconfiguration. The proposed framework is implemented as a standalone MATLAB/Simulink library to facilitate time-domain simulation and to provide a structured basis for future eigenvalue-based small-signal stability studies. The approach is validated on a point-to-point HVDC system and benchmarked against an EMT model. Results demonstrate that the DQsym-based implementation achieves EMT-level fidelity while maintaining state-space structure suitable for scalable, system-level analysis of converter-dominated grids.

The growing High-Voltage Direct Current transmission networks require modern control strategies in converter stations to ensure reliable operation and uninterrupted energy supply, particularly under unstable and low Short-Circuit ratio conditions. Conventional Grid Following converters become unstable in low Short-Circuit ratio scenarios, while modern Grid Forming converters, though more robust, exhibit slower dynamic response in high Short-Circuit ratio scenarios. This paper presents a hybrid control switching strategy based on polytopic Lyapunov functions that combines the strengths of Grid Following Control and Grid Forming Control strategies. Switching between these control strategies occurs at defined hyperplanes of polytopes derived from the state-space equations, enabling the system to maintain fast and stable performance under changing Short-Circuit ratio conditions of the grid. Because the method is grounded in polytopic Lyapunov function theory, it demonstrates inherent large-signal stability. This proposed Hybrid Control Strategy is validated using real-time ^ő simulations, showing robust performance during Short-Circuit ratio variations, highlighting its potential for future High-Voltage Direct Current systems.

This paper proposes the use of Model Predictive Control (MPC) with an exponential cost function for the Modular Multilevel Converter (MMC), which is widely recognized as a preferred converter topology for integrating and converting renewable energy sources into electrical energy. MPC provides a superior control strategy in the presence of system constraints, a straightforward control design, facilitates the inclusion of multiple control objectives through a flexible cost-function formulation, and offers excellent control performance. By formulating an appropriate cost function, an MMC’s operational goals can be effectively achieved through MPC. However, non-exponential MPC approaches typically employ a rectangular moving-horizon window whose length matches the chosen prediction horizon, which can affect closed-loop stability. The results based on the non-exponential cost further reveal that the choice of prediction horizon notably influences the numerical conditioning of MPC algorithms. In particular, as the prediction horizon lengthens, the numerical condition tends to degrade rapidly when a large control horizon is used. This research work uses an exponentially weighted moving horizon window to overcome these issues. Employing the exponential-based cost function further significantly reduces the condition number of the Hessian matrix, thereby improving the numerical properties of the MPC. We further analyze the effects of different constraints, observing that the MPC strictly adheres to them and that the control variable influences the response of the MMC plant’s performance. We further compared our results with those of other controllers and analyzed performance metrics, demonstrating that the exponential MPC is effective in this case. Additionally, the results presented in this paper demonstrate the prescribed degree of stability and highlight the importance of fine-tuning key MPC parameters for the MMC model. The exponential-based MPC is validated for the MMC under scenarios involving small and large active and reactive power disturbances, considering offline simulations.

With the growing integration of Modular Multilevel Converters (MMCs) in Multi-Terminal Direct Current (MTDC) transmission systems, there is a growing need for control strategies that balance economic efficiency with robust dynamic performance. This paper presents an enhanced Optimal Power Flow (OPF)-based framework for hybrid AC-MTDC systems, incorporating a novel droop control strategy that jointly coordinates DC-voltage and AC-frequency regulation. By embedding frequency control loops into the MMCs, the method enables system-wide coordination that enhances power sharing and improves resilience under disturbances. The proposed strategy dynamically adjusts converter operating points to minimize generation costs and DC-voltage deviations, balancing economic objectives with system stability. A modified Nordic test system integrated with a four-terminal MTDC grid is used to validate the approach. Optimization is performed using Julia, while the system's dynamic performance is evaluated through electromagnetic transient simulations with the EMTP software. Case studies across multiple scenarios demonstrate that the proposed droop control achieves markedly improved frequency and voltage robustness over active power control, while incurring lower generation costs than the adaptive droop benchmark. The results highlight the ability of the proposed strategy to deliver cost-effective operation without compromising performance, offering a promising solution for the coordinated control of future hybrid AC-DC transmission networks.

The rapid expansion of offshore wind and solar is placing increasing pressure on transmission infrastructure, challenging energy security, sustainability, and affordability. In response, global policy initiatives are advancing high-voltage direct current (HVDC) "express energy highways"to efficiently transfer bulk renewable power. Voltage-source converter (VSC)-based multi-Terminal HVDC (MT-HVDC) grids offer a scalable offshore solution but require advanced control to manage their fast nonlinear dynamics and low inertia. This paper presents a real-Time control and dispatch framework for a five-Terminal radial-mesh VSC-MMC HVDC grid. A model predictive control (MPC) strategy is developed, automatically coded from Simulink, and deployed on a real-Time target for controller-hardware-in-The loop (CHIL) validation using an RTDS® simulator. The setup includes dynamic braking resistors, DC breakers, and integrated human machine and dispatch interfaces for real-Time tuning and supervisory control. Comparative testing under power setpoint changes and AC/DC fault scenarios shows that the proposed MPC achieves more accurate active and reactive power tracking with lower overshoot, improves damping of post-fault oscillations, and enables seamless coordination between local and system-level operations. These results confirm the practical feasibility and operational resilience of MPC-based control for future offshore MT-HVDC systems.

This paper describes an advanced algorithm for current control in a modular multilevel power converter. The proposed algorithm is based on deadbeat model predictive control, which is further enhanced by applying an artificial neural network. Rather than supplanting the existing predictive controller, the neural network is used to supplement it, thereby improving control performance. The reference current tracking capability is enhanced under all operating conditions, particularly in cases where the reactive elements of the converter exhibit a relatively high resistive component. By introducing the neural network, the inherent delay of model predictive control in such scenarios is virtually eliminated, which reduces the overall control error. The network is trained offline on a set of control inputs obtained by optimizing the current response in a wide range of operating modes. Algorithm verification is performed for a three-phase modular multilevel converter in different steady-state and transient conditions. A state-of-the-art high-fidelity real-time simulator is used in both the algorithm development and verification stages.

The displacement of synchronous generation by converter-interfaced resources is reshaping power system dynamics and has made resonance stability, a category newly recognized in modern stability framework, an increasingly pressing concern. Compared with converter-driven stability, however, resonance stability has received limited systematic treatment and still lacks a unifying perspective that connects its diverse manifestations. This paper revisits resonance from a unified, mechanism-oriented viewpoint, interpreting it as an oscillatory phenomenon arising from strong coupling among closely spaced mechanical, electrical, electromechanical, or control modes of different subsystems or control loops. On this basis, resonance is classified into four types according to the interacting modal pairs, thereby placing classical subsynchronous resonance, electromechanical modal resonance, and emerging converter-related resonances within a single framework. Representative incidents are first reviewed to show how the dominant patterns have evolved from synchronous-machine interactions toward broadband, converter-dominated ones. The main analysis methods are then surveyed and mapped to the resonance types they best address. Finally, mitigation strategies are organized by resonance type around three basic levers: damping injection, impedance reshaping, and modal detuning. The review aims to provide a coherent basis for interpreting mechanisms, selecting methods, and designing mitigation measures in converter-dominated power systems.

The increasing integration of renewable energy, particularly offshore wind, introduces significant uncertainty into hybrid AC-HVDC systems due to forecast errors and power fluctuations. Conventional control strategies typically rely on fixed setpoints and neglect frequency deviations, which compromise system stability under rapid renewable variations. To address the challenge, this paper presents an optimal power flow (OPF)-based adaptive control framework for hybrid AC-MTDC systems with offshore wind integration. To enable the time-coupled OPF to produce anticipatory setpoints under wind variability, a Random Forest-based wind-speed forecast is integrated as an uncertainty-aware data source, providing the OPF with an informed view of the wind trajectory over the dispatch horizon. The resulting baseline setpoints are further adjusted in real time through an adaptive droop control scheme that simultaneously regulates DC voltage and AC frequency to enhance robustness across a broad class of disturbances. The effectiveness of the proposed framework is validated through hardware-in-the-loop simulations against three benchmark control modes, demonstrating consistent and balanced performance under wind-forecast uncertainty, AC-side load steps, and DC-side faults.

Modern power systems with high penetration of inverter-based resources (IBRs) exhibit fast dynamics and complex harmonic interactions that challenge conventional modelling tools. Electromagnetic transient (EMT) simulations provide high fidelity but are computationally demanding for large-scale studies due to small time-step requirements, whereas conventional phasor-domain models neglect harmonic and higher-frequency effects to allow for larger time-steps. This paper proposes a unified dynamic-phasor-based framework (DQsym), implemented as a MATLAB/Simulink library, that combines dynamic phasors with multiple rotating reference frames and defines explicit algebraic rules for harmonic-domain operations compatible with state-space formulations, enabling systematic assembly of interconnected system-level models beyond isolated component representations. The formulation supports modelling across multiple harmonic orders and is expressed in state space, providing a natural pathway for future integration with small-signal analysis and control design tools, although such extensions are outside the scope of this paper. The approach is validated through: 1) benchmark case demonstrating higher-order harmonic modelling capability and 2) simulations of an IEEE 9-bus system expanded with point-to-point HVDC transmission based on a modular multilevel converter (MMC), where the framework reproduces fundamental and second-harmonic dynamics indicating that DQsym reproduces the overall harmonic pattern and closely matches the fundamental component compared with EMT results. The proposed framework provides a structured and accurate harmonic-domain modelling tool for the analysis of IBR-rich power systems.

Hybrid AC/voltage source converter-based multi-terminal DC (VSC-MTDC) power grids play a crucial role in enabling long-distance power transmission and flexible interconnection between AC grids. To fully leverage the functional advantages of such systems, it is essential that they operate in or close to optimal power flow (OPF) conditions. To address this, ACDC-OpFlow is developed as an open-source and cross-language framework for solving AC/DC OPF problems. Its core innovation lies in a unified modeling structure that supports MATLAB, Python, Julia, and C++, with Gurobi used as a consistent solver backend. This framework is beginner-friendly and allows users to work in their preferred programming languages. Both text-based and graph-topology results are provided to help users understand the system-wide power flow distribution and operational status. This work presents the design concept of ACDC-OpFlow, showcases representative example results, and discusses the performance differences observed in multiple programming language implementations.

The potential of advanced neural networks (NNs) has yet to be explored in the field of HVDC transmission. Implementing such intelligent computational techniques on a real-time digital simulator (RTDS) is challenging due to the need for rapid computation and the risk of overfitting with extensive data generated at tiny time steps. To overcome these limitations, different NN techniques are studied using a supervised and reinforced imitation learning method to mimic the suggested controller with labeled data for real-time applications. Furthermore, the NN component does not necessarily just take a label, and therefore, the authors propose a more advanced approach by incorporating reinforced learning through an error-tracking mechanism into the NN, apart from its loss function. The initial offline processing identifies the best-suited NN technique for online computational feasibility. Both online and offline training methods as well as online adjustments are showcased to provide a robust control solution that is easy to implement. This work deals with developing an intuitive and versatile Toolbox installed on a real-time simulator platform that can integrate complex NN-based control strategies. Extensive simulations on the RTDS platform and experimental investigations of the four terminal HVDC systems validate the interest and viability of the proposed design methodology.

The modular multilevel converter (MMC) uses many power electronic components in the high voltage direct current (HVDC) application. One of the major concerns in half-bridge MMC is the fault in the converter submodules. It raises the question of whether the reliability and high-quality performance of the MMC can be increased significantly as the active device controls the power flow between the AC- and DC-sides. During the SM fault within the MMC leg, the unbalance is introduced in-side the MMC converter. The unbalanced voltage within the leg of the MMC will continuously introduce an AC-current component on the DC-side of the converter. Thus, the hybrid proportional-internal (PI) control and proportional-resonant control (PR) is introduced in controlling the power flow within the internal MMC to eliminate the AC-current component and ensure pure DC-current in the internal MMC. This study investigates the internal power flow control of a three-phase rectifier MMC with symmetric and asymmetric SM fault conditions. Compared with conventional control methods, the proposed control can tolerate SM faults and eliminate the AC-current component within the converter, increasing the converter's performance. Simulation results are included and discussed to verify the proposed control.

There is a lack of knowledge on the development of a controller model into a black box dynamically-linked library (DLL) file which can be flexibly tested with a power system model through a real-time co-simulation platform. Developing a test bed to demonstrate the performance and compliance analysis enhances the approval of novel control and protection strategies. This paper presents the development of a controller DLL and a test bench that integrates this DLL with a real-time simulator. The DLL contains a previously developed grid-forming controller for the type‑3 wind turbine and works in co-simulation with the power system model on Real-Time Digital Simulator (RTDS) in real-time.

The increased use of High-Voltage Direct Current transmission networks requires appropriate control strategies for the converter stations, which are crucial to ensure uninterrupted energy supply. In this paper, a Polytopic Lyapunov Function-based Hybrid switching control strategy is implemented to combine the merits of Grid Following and Grid Forming control strategies by switching alternatively from one to another at the polytopes’ hyperplanes to ensure good system response even for faulty conditions. The state space equations of the control strategies are used to form the state hyperplanes for the switching rule. Since the hybrid switching control is based on the Polytopic Lyapunov Function, the system is inherently Large Signal Stable. The results obtained by real-time-based simulations using RTDS verify the designed control for various transient phenomena.

The Modular Multilevel Converter (MMC) has garnered significant interest recently due to its superior harmonic performance and improved efficiency in high-voltage direct current electrical grids. Model Predictive Control (MPC) is widely adopted for the MMC applications, as it provides a straightforward control design, facilitates the inclusion of multiple control objectives through a flexible cost function formulation, and offers excellent control performance. An emerging and promising solution involves integrating MPC with machine learning (ML)-based models, in which neural networks learn MPC behavior and predict the results as the traditional MPC does.In this paper, a multi-layered neural network is designed to approximate the control behavior of MPC correctly, enabling a substantial reduction in computational effort during real-time operation and replacing the complex optimization routines of MPC with lightweight neural network regression models that are both efficient and decoupled from the algorithmic complexity of traditional MPC. The performance of controllers is evaluated under both small and large disturbances in active power and reactive power.

With the domination of modular multilevel converters (MMCs) interfaced power grids, especially for transmission of the wind generated energy, the control of such power electronic interfaced grids is of an utmost important for the proper operation and grid stability. This control is very complex due to multivariable intercoupling and plausible nonlinearity. To enhance the grid stability and reduce the total harmonic distortion (THD) of the converter, the paper proposes development of an optimal voltage level-model predictive control (OVL-MPC) for a fast dynamic response, integrated with classical proportional–integral (PI) outer-loop control for robust steady-state performance. This control eliminates the problems of poor steady-state performance of MPC while achieving faster transient response in comparison to the classical proportional integral (PI) dual-loop control. The work proposes OVL-MPC for lower computational burden in comparison to switching state-based MPC, for the inner loop replacing the classical PI inner loop. With the inherent advantages of lower computational burden and superior transient performance, AC current deadbeat controller is used for the modulation in OVL-MPC. To improve the robustness of the control method, the Moore–Penrose pseudo-inversion is applied to address control parameter mismatches, while the Smith predictor compensates for time delays. The designed control algorithm is tested with two real-time simulation platforms, i.e., OPAL-RT and RTDS for thorough power system validation.

The deployment of voltage source converters (VSC) to facilitate flexible interconnections between the AC grid, renewable energy system (RES) and Multi-terminal DC (MTDC) grid is on the rise. However, significant challenges exist in exploiting coordinated operations for such AC/VSC-MTDC hybrid power systems. One of the most critical issues is how to achieve the optimal operation of such wide-area systems involving several power entities with as minimal communication burden as possible. To address this issue, an enhanced AC/DC optimal power flow (OPF) is specifically proposed. Firstly, a mixed-integer convex AC/DC OPF model is explicitly formulated to describe the optimal operation of such hybrid power systems. Subsequently, a nested distributed optimization method with double iteration loops is developed to offer optimal system-wide decision-making through a more “thorough” distributed communication architecture. In the outer iteration, the original AC/DC OPF problem is decomposed into several slave problems (SPs) associated with systems (including the AC grid and RESs) and one master problem (MP) associated with the integrated VSC-MTDC grid. Generalized Benders decomposition (GBD) serves to solve the master and slave problems iteratively. Techniques such as multi-cut generation and asynchronous updating are utilized to upgrade the GBD performance of computation efficiency and address communication delays. In the inner iteration, the master problem is continuously decomposed into multiple sub-MPs associated with individual VSCs. The alternating direction method of multipliers (ADMM) is employed to solve these sub-MPs iteratively. Proximal terms and heuristic approaches are embedded to enable parallel computation and handling of integer variables. Numerical experiment results finally validate the effectiveness of the proposed enhanced AC/DC OPF. The constructed AC/DC OPF model exhibits acceptable accuracy in terms of power flow calculation, and the developed nested distributed optimization method showcases decent convergence rate and solution optimality performances.

DC fault location technology is crucial for estimating the fault location and developing multi-terminal direct current (MTDC) systems. This article presents a novel fault location method using the parameter fitting approach. The propagation of traveling waves (TWs) in the decoupled line-mode fault network is first discussed, resulting in analytical expressions for the backward line-mode current TWs containing fault location information. Then, the adaptive multi-step Levenberg–Marquardt (AMLM) algorithm is applied for parameter fitting owing to its fast processing speed and precision. The exact fault location is estimated using the fitted coefficient. Different testing MTDC systems modeled in PSCAD/EMTDC and a real-time digital simulator (RTDS) validate the proposed fault location method. Based on numerous simulation tests, the AMLM-based parameter fitting and the proposed method are accurate, with errors smaller than 0.5%. Compared to the existing methods, the proposed method has desired performance under close-in faults, can withstand 35 dB noise interference, and obviates the need for an extremely high sampling frequency, estimation of tws velocity, and communication devices.