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Journal articleMiller LM, Elliott ADT, Mitcheson PD, et al., 2016,
This paper presents a complete optimization of a piezoelectric vibration energy harvesting system, including a piezoelectric transducer, a power conditioning circuit with full semiconductor device models, a battery and passive components. To the authors awareness, this is the first time and all of these elements have been integrated into one optimization. The optimization is done within a framework, which models the combined mechanical and electrical elements of a complete piezoelectric vibration energy harvesting system. To realize the optimization, an optimal electrical damping is achieved using a single-supply pre-biasing circuit with a buck converter to charge the battery. The model is implemented in MATLAB and verified in SPICE. The results of the full system model are used to find the mechanical and electrical system parameters required to maximize the power output. The model, therefore, yields the upper bound of the output power and the system effectiveness of complete piezoelectric energy harvesting systems and, hence, provides both a benchmark for assessing the effectiveness of existing harvesters and a framework to design the optimized harvesters. It is also shown that the increased acceleration does not always result in increased power generation as a larger damping force is required, forcing a geometry change of the harvester to avoid exceeding the piezoelectric breakdown voltage. Similarly, increasing available volume may not result in the increased power generation because of the difficulty of resonating the beam at certain frequencies whilst utilizing the entire volume. A maximum system effectiveness of 48% is shown to be achievable at 100 Hz for a 3.38-cm3 generator.
Journal articleKunjumuhammed LP, Pal BC, Oates C, et al., 2016,
Large offshore wind farms are usually composed ofseveral hundred individual wind turbines, each turbine havingits own complex set of dynamics. The analysis of the dynamicinteraction between wind turbine generators (WTG), interconnectingac cables, and voltage source converter (VSC) based HighVoltage DC (HVDC) system is difficult because of the complexityand the scale of the entire system. The detailed modelling andmodal analysis of a representative wind farm system reveal thepresence of several critical resonant modes within the system.Several of these modes have frequencies close to harmonicsof the power system frequency with poor damping. From acomputational perspective the aggregation of the physical modelis necessary in order to reduce the degree of complexity to apractical level. This paper focuses on the present practices ofthe aggregation of the WTGs and the collection system, andtheir influence on the damping and frequency characteristics ofthe critical oscillatory modes. The effect of aggregation on thecritical modes are discussed using modal analysis and dynamicsimulation. The adequacy of aggregation method is discussed.
Journal articleForni P, Angeli D, 2016,
The ISS approach to the stability and robustness properties of nonautonomous systems with decomposable invariant sets: an overview, European Journal of Control, Vol: 30, Pages: 50-60, ISSN: 0947-3580
This paper is an overview of recent developments in the Input-to-State Stability framework, dealing in particular with the extension of the classical concept to systems with multiple invariant sets and possibly evolving on Riemannian manifolds. Lyapunov-based characterizations of the properties are discussed as well as applications to the study of cascaded nonlinear systems.
Journal articleGuo J, Liang J, Zhang X, et al., 2016,
The half-bridge-based modular multilevel converter (MMC) has emerged as the favored converter topology for voltage-source HVDC applications. The submodules within the converter can be constructed with either individual insulated-gate bipolar transistor (IGBT) modules or with series-connected IGBTs, which allows for different redundancy strategies to be employed. The main contribution of this paper is that an analytical method was proposed to analyze the reliability of MMCs with the consideration of submodule arrangements and redundancy strategies. Based on the analytical method, the relative merits of two approaches to adding redundancy, and variants created by varying the submodule voltage, are assessed in terms of overall converter reliability. Case studies were conducted to compare the reliability characteristics of converters constructed using the two submodule topologies. It is found that reliability of the MMC with series-connected IGBTs is higher for the first few years but then decreases rapidly. By assigning a reduced nominal voltage to the series valve submodule upon IGBT module failure, the need to install redundant submodules is greatly reduced.
Journal articleAngeli D, Casavola A, Tedesco F, 2016,
Economic Model Predictive Control is a technique for optimization of economic revenues arising from controlled dynamical processes that has established itself as a variant of standard Tracking Model Predictive Control. It departs from the latter in that arbitrary cost functions are allowed in the formulation of the stage cost. This paper takes a further step in expanding the applicability of Economic Model Predictive Control by illustrating how the paradigm can be adapted in order to accommodate time-varying or parameter-varying costs.
Journal articleBettiol P, Vinter RB, 2016,
L∞ estimates on trajectories confined to a closed subset, for control systems with bounded time variation, Mathematical Programming, Vol: 168, Pages: 201-228, ISSN: 1436-4646
The term ‘distance estimate’ for state constrained control systems refers to an estimate on the distance of an arbitrary state trajectory from the subset of state trajectories that satisfy a given state constraint. Distance estimates have found widespread application in state constrained optimal control. They have been used to establish regularity properties of the value function, to establish the non-degeneracy of first order conditions of optimality, and to validate the characterization of the value function as a unique solution of the HJB equation. The most extensively applied estimates of this nature are so-called linear L∞L∞ distance estimates. The earliest estimates of this nature were derived under hypotheses that required the multifunctions, or controlled differential equations, describing the dynamic constraint, to be locally Lipschitz continuous w.r.t. the time variable. Recently, it has been shown that the Lipschitz continuity hypothesis can be weakened to a one-sided absolute continuity hypothesis. This paper provides new, less restrictive, hypotheses on the time-dependence of the dynamic constraint, under which linear L∞L∞ estimates are valid. Here, one-sided absolute continuity is replaced by the requirement of one-sided bounded variation. This refinement of hypotheses is significant because it makes possible the application of analytical techniques based on distance estimates to important, new classes of discontinuous systems including some hybrid control systems. A number of examples are investigated showing that, for control systems that do not have bounded variation w.r.t. time, the desired estimates are not in general valid, and thereby illustrating the important role of the bounded variation hypothesis in distance estimate analysis.
Journal articleChen B, Pin G, Ng WM, et al., 2016,
A fast-convergent modulation integral observer for online detection of the fundamental and harmonics in grid-connected power electronics systems, IEEE Transactions on Power Electronics, Vol: 32, Pages: 2596-2607, ISSN: 1941-0107
Harmonics detection is a critical element of activepower filters. A previous review has shown that the RecursiveDiscrete Fourier Transform and the Instantaneous p-q Theory areeffective solutions to extracting power harmonics in single-phaseand three-phase power systems, respectively. This paper presentsthe operating principle of a new modulation function integralobserver algorithm that offers a fast solution for the extraction ofthe fundamental current and the total harmonic current whencompared with existing methods. The proposed method can beapplied to both single- and three-phase systems. The observerbasedalgorithm has an advantageous feature of being able to betuned offline for a specific application, having fast convergenceand producing estimated fundamental component with highcircularity. It has been tested with both simulations and practicalmeasurements for extracting the total harmonic current in ahighly efficient manner. The results have confirmed that theproposed tool offers a new and highly effective alternative to thesmart grid industry.
Conference paperLi S, Lee ATL, Siew-Chong-Tan, et al., 2016,
In this paper, a plug-and-play ripple mitigation technique is proposed. It requires only the sensing of the DC-link voltage and can operate fully independently to remove the low-frequency voltage ripple. The proposed technique is nonintrusive to the existing hardware and enables hot-swap operation without disrupting the normal functionality of the existing power system. It is user-friendly, modular and suitable for plug-and-play operation. The experimental results demonstrate the effectiveness of the ripple-mitigation capability of the proposed device. The DC-link voltage ripple in a 110 W miniature hybrid system comprising an AC/DC converter and two resistive loads is shown to be significantly reduced from 61 V to only 3.3 V. Moreover, it is shown that with the proposed device, the system reliability has been improved by alleviating the components' thermal stresses.
Conference paperYang Y, Tan SC, Hui SY, 2016,
An adaptive reference model predictive control (ARMPC) approach is proposed as an alternative means of controlling power converters in response to the issue of steady-state residual errors presented in power converters under the conventional model predictive control (MPC). Differing from other methods of eliminating steady-state errors of MPC based control, such as MPC with integrator, the proposed ARMPC is designed to track the so-called virtual references instead of the actual references. Subsequently, additional tuning is not required for different operating conditions. In this paper, ARMPC is applied to a single-phase full-bridge voltage source inverter (VSI). It is experimentally validated that ARMPC exhibits strength in substantially eliminating the residual errors in environment of model mismatch, load change, and input voltage change, which would otherwise be present under MPC control. Moreover, it is experimentally demonstrated that the proposed ARMPC shows a consistent erasion of steady-state errors, while the MPC with integrator performs inconsistently for different cases of model mismatch after a fixed tuning of the weighting factor.
Conference paperAldhaher S, Mitcheson PD, Yates DC, 2016,
This paper will present the modelling, analysis and design of a load-independent Class EF inverter. This inverter is able to maintain zero-voltage switching (ZVS) operation and produce a constant output current for any load value without the need for tuning or replacement of components. The load-independent feature of this inverter is beneficial when used as the primary coil driver in multi megahertz high power inductive wireless power transfer (WPT) applications where the distance between the coils and the load are variable. The work here begins with the traditional load-dependent Class EF topology for inversion and then specifies the criteria that are required to be met in order achieve load-independence. The design and development of a 240W load-independent Class EF inverter to drive the primary coil of a 6.78MHz WPT system will be discussed and experimental results will be presented to show the load-independence feature when the distance between the coils of the WPT system changes.
Journal articleScarciotti G, 2016,
Low computational complexity model reduction of power systems with preservation of physical characteristics, IEEE Transactions on Power Systems, Vol: 32, Pages: 743-752, ISSN: 1558-0679
A data-driven algorithm recently proposed to solvethe problem of model reduction by moment matching is extendedto multi-input, multi-output systems. The algorithm isexploited for the model reduction of large-scale interconnectedpower systems and it offers, simultaneously, a low computationalcomplexity approximation of the moments and the possibilityto easily enforce constraints on the reduced order model. Thisadvantage is used to preserve selected slow and poorly dampedmodes. The preservation of these modes has been shown to beimportant from a physical point of view and in obtaining anoverall good approximation. The problem of the choice of the socalledtangential directions is also analyzed. The algorithm andthe resulting reduced order model are validated with the studyof the dynamic response of the NETS-NYPS benchmark system(68-Bus, 16-Machine, 5-Area) to multiple fault scenarios.
Journal articleMoreira R, Moreno R, Strbac G, 2016,
Energy storage can provide services to several sectors in electricity industry, including generation, transmission and distribution, where conflicts and synergies may arise when storage is used to manage network congestion and provide services in energy and balancing markets. In this context, this study proposes an optimisation model to coordinate multiple services delivered to various market participants that uses corrective actions to resolve conflicts between provision of distribution network services (e.g. congestion and security of supply) and other services. The model maximises storage profit by scheduling active and reactive power to provide portfolio of services including distribution network congestion management, energy price arbitrage, frequency response and reserve services remunerated at different prices. The authors demonstrate that adopting corrective security to provide network services and deal with network congestion in a post-fault fashion, is overall more beneficial despite the energy needed to be stored during pre-fault conditions for applying post-contingency actions right after a network fault occurs. Furthermore, the authors' analysis shows that application of corrective security can benefit both (i) storage owners through increased revenues in energy and balancing services markets and (ii) Distribution Network Operators through reduction in payments to storage owners and increased utilisation of network infrastructure.
Journal articleBeddard A, sheridan CE, Barnes M, et al., 2016,
Modular multilevel converters (MMCs) have become the converter topology of choice for voltage-source converter-high-voltage direct-current systems. Excellent work has previously been conducted to develop much needed average value models (AVM) for these complex converters; however, there a number of limitations as highlighted in this paper. This paper builds on the existing models, proposing numerous modifications and resulting in an enhanced MMC-AVM, which is significantly more accurate and which can be used for a wider range of studies, including DC faults.
Journal articleCanizares C, Farnandes T, Gerladi E, et al., 2016,
This paper summarizes a set of six benchmark systemsfor the analysis and control of electromechanical oscillationsin power systems recommended by the IEEE Task Force onBenchmark Systems for Stability Controls of the Power SystemDynamic Performance Committee. The benchmark systems werechosen for their tutorial value and particular characteristicsleading to control system design problems relevant to the researchcommunity. For each benchmark, the modelling guidelinesare provided, along with eigenvalues and time-domain resultsproduced with at least two simulation software, and onepossible control approach is provided for each system as well.Researchers and practicing engineers are encouraged to use thesebenchmark systems when assessing new oscillation dampingcontrol strategies.
Conference paperDe Paola A, Angeli D, Strbac G, 2016,
This paper characterizes optimal control policies for wind farms operated as frequency response services in case of a fault of conventional generators. The frequency support is provided through temporary over-production: when frequency drops, the turbines move from the steady-state operating point and extra power is produced by slowing down the turbines and releasing part of their kinetic energy. The control task is formulated and solved as an optimal containment problem: the time during which an extra quantity of power can be produced, within the set speed constraints for each turbine, is maximized. The solutions are calculated and compared for different assumptions on the electric torque of the turbines.
Journal articleNewbery D, Strbac G, Viehoff I, 2016,
The European Commission's Target Electricity Model (TEM) aims to integrate EU electricity markets. This paper estimates the potential benefit of coupling interconnectors to increase the efficiency of trading day-ahead, intra-day and balancing services across borders. Further gains are possible by eliminating unscheduled flows and avoiding the curtailment of renewables with better market design. In the short run the gains could be as high as €3.9 billion/yr, more than 100% of the current gains from trade. About one-quarter of this total comes from day-ahead coupling and another third from shared balancing. If shared balancing is so valuable, completing the TEM becomes more urgent, and regulators should ensure these gains are paid to interconnectors to make the needed investment in the cross-border links more commercially profitable.
Conference paperGiannelos S, Konstantelos I, Strbac G, 2016,
Under the current EU legislation, Distribution NetworkOperators (DNOs) are expected to provide firm connections to newDG, whose penetration is set to increase worldwide creating theneed for significant investments to enhance network capacity.However, the uncertainty around the magnitude, location andtiming of future DG capacity renders planners unable to accuratelydetermine in advance where network violations may occur. Hence,conventional network reinforcements run the risk of assetstranding, leading to increased integration costs. A novel stochasticplanning model is proposed that includes generalized formulationsfor investment in conventional and smart grid assets such asDemand-Side Response (DSR), Coordinated Voltage Control (CVC)and Soft Open Point (SOP) allowing the quantification of theiroption value. We also show that deterministic planning approachesmay underestimate or completely ignore smart technologies.
Journal articleChakravorty D, Chaudhuri B, Hui SYR, 2016,
Flexibility in certain types of loads could be exploited to provide fast and controllable power reserve if the supply voltage/frequency is controlled using existing power electronic interfaces (e.g. motor drives) or additional ones like recently proposed Electric Springs. Such a load together with its power electronic interface forms a so called ‘smart load’. Effectiveness of static smart loads for primary frequency response provision has been shown in previous papers through case studies ona segment of the LV/MV distribution network. In this paper, collective contribution of both static and motor type smart loads to rapid frequency response provision is demonstrated through a case study on the Great Britain (GB) transmission system. The active power reserve available from such smart loads are quantified and aggregated at each node at the transmission level (275/400 kV). The study shows that the smart loads collectively offer a short-term power reserve which is comparable to the spinning reserve in the GB system and thus can ensure acceptable frequency deviation and its rate of change (RoCoF) following a large infeed loss.
Conference paperspallarossa C, merlin M, Green TC, 2016,
During the recent years, the number of High Voltage Direct Current (HVDC) links, used for accommodating renewable energy sources and for interconnecting different power systems, has increased significantly. Apart from enhancing system stability, HVDC links can be used to provide ancillary services, such as frequency support. This paper investigates the potential of the energy storage capability of Modular Multi-level Converters (MMCs) to contribute to the frequency response. MMCs provide fast released energy through their cells capacitor that may be used to improve the system inertial response during the fault transient after a loss of generation event. To assess the effect of the converter energy storage capability, a mixed AC and DC transmission platform is developed in PowerFactory. It consists of a multi-machine system including a point-to-point MMC based HVDC link. A generation outage is applied to investigate the contribution of the MMC energy storage capability to the frequency recovery.
Journal articleTeng F, Strbac G, 2016,
High penetration of wind generation causes concerns over frequency stability, as currently wind plants do not provide frequency response support. Extensive research has been conducted to investigate alternative designs of controllers to facilitate the provision of synthetic inertia and primary frequency response from wind plants. However, frequency response support from wind plants differs from that provided by conventional plants and its impact on the system's economic performance is not yet fully understood. In this context, this paper develops a novel methodology to incorporate the frequency response support from wind plants into generation scheduling, thus enabling the benefits of alternative control strategies to be quantified. Studies are carried out on the future Great Britain power system with different wind energy penetration levels and frequency response requirements. The impact of the uncertainty associated with the quantity of wind plants being online and the energy recovery effect are also analyzed. The results demonstrate that the benefits of frequency response support from wind plants may be significant, although these are system specific. The proposed model could also inform the development of grid codes, market mechanisms, and business cases associated with the frequency response support from wind plants.
Journal articleGoverdovsky V, Yates DC, Willerton M, et al., 2016,
A fully synchronized modular multichannel software-defined radio (SDR) testbed has been developed for the rapid prototyping and evaluation of array processing algorithms. Based on multiple universal software radio peripherals, this testbed is low cost, wideband, and highly reconfigurable. The testbed can be used to develop new techniques and algorithms in a variety of areas including, but not limited to, direction finding, source triangulation, and wireless sensor networks. A combination of hardware and software techniques is presented, which is shown to successfully remove the inherent phase and frequency uncertainties that exist between the individual SDR peripherals. The adequacy of the developed techniques is demonstrated through the application of the testbed to super-resolution direction finding algorithms, which rely on accurate phase synchronization.
Journal articleAldhaher S, Mitcheson PD, Yates DC, 2016,
Design and Development of a Class EF<sub>2</sub> Inverter and Rectifier for Multi-megahertz Wireless Power Transfer Systems, IEEE Transactions on Power Electronics, Vol: 31, Pages: 8138-8150, ISSN: 1941-0107
This paper presents the design and implementation of a Class EF2 inverter and Class EF2 rectifier for two -W wireless power transfer (WPT) systems, one operating at 6.78 MHz and the other at 27.12 MHz. It will be shown that the Class EF2 circuits can be designed to have beneficial features for WPT applications such as reduced second-harmonic component and lower total harmonic distortion, higher power-output capability, reduction in magnetic core requirements and operation at higher frequencies in rectification compared to other circuit topologies. A model will first be presented to analyze the circuits and to derive values of its components to achieve optimum switching operation. Additional analysis regarding harmonic content, magnetic core requirements and open-circuit protection will also be performed. The design and implementation process of the two Class-EF2-based WPT systems will be discussed and compared to an equivalent Class-E-based WPT system. Experimental results will be provided to confirm validity of the analysis. A dc-dc efficiency of 75% was achieved with Class-EF2-based systems.
Conference paperScarciotti G, Astolfi A, 2016,
Moments at "discontinuous signals" with applications: model reduction for hybrid systems and phasor transform for switching circuits, 22nd International Symposium on Mathematical Theory of Networks and Systems
We provide an overview of the theory and applicationsof the notion of moment at “discontinuous interpolationsignals”, i.e. the moments of a system for input signals thatdo not satisfy a differential equation. After introducing thetheoretical framework, which makes use of an integral matrixequation in place of a Sylvester equation, we discuss someapplications: the model reduction problem for linear systems atdiscontinuous signals, the model reduction problem for hybridsystems and the discontinuous phasor transform for the analysisof circuits powered by discontinuous sources.
Journal articleJiang J, Di Franco P, Astolfi A, 2016,
This paper presents shared-control algorithms for the kinematic and dynamic models of a mobile robot with a feasible configuration set defined by means of linear inequalities. The shared-control laws based on a hysteresis switch are designed in the case in which absolute positions are not available. Instead, we measure the distances to obstacles and angular differences. Formal properties of the closed-loop systems with the shared control are established by a Lyapunov-like analysis. Simulation results and experimental results are presented to show the effectiveness of the algorithm.
Journal articleMohtashami S, Pudjianto D, Strbac G, 2016,
This paper presents a multiyear distribution network planning optimization model for managing the operation and capacity of distribution systems with significant penetration of distributed generation (DG). The model considers investment in both traditional network and smart grid technologies, including dynamic line rating, quadrature-booster, and active network management, while optimizing the settings of network control devices and, if necessary, the curtailment of DG output taking into account its network access arrangement (firm or non-firm). A set of studies on a 33 kV real distribution network in the U.K. has been carried out to test the model. The main objective of the studies is to evaluate and compare the performance of different investment approaches, i.e., incremental and strategic investment. The studies also demonstrate the ability of the model to determine the optimal DG connection points to reduce the overall system cost. The results of the studies are discussed in this paper.
Journal articleWu FF, Varaiya PP, Hui RSY, 2016,
A future smart grid must fulfill the vision of the Energy Internet in which millions of people produce their own energy from renewables in their homes, offices, and factories and share it with each other. Electric vehicles and local energy storage will be widely deployed. Internet technology will be utilized to transform the power grid into an energy-sharing inter-grid. To prepare for the future, a smart grid with intelligent periphery, or smart GRIP, is proposed. The building blocks of GRIP architecture are called clusters and include an energy-management system (EMS)-controlled transmission grid in the core and distribution grids, micro-grids, and smart buildings and homes on the periphery; all of which are hierarchically structured. The layered architecture of GRIP allows a seamless transition from the present to the future and plug-and-play interoperability. The basic functions of a cluster consist of ① dispatch, ② smoothing, and ③ mitigation. A risk-limiting dispatch methodology is presented; a new device, called the electric spring, is developed for smoothing out fluctuations in periphery clusters; and means to mitigate failures are discussed.
Journal articleMok KT, Wang MH, Tan SC, et al., 2016,
There is a growing interest in using dc power systems and microgrids for our electricity transmission and distribution, particularly with the increasing penetration of photovoltaic power systems. This paper presents an electric active suspension technology known as the dc electric springs (DC-ES) for voltage stabilization and power quality improvement. The basic operating modes and characteristic of a DC-ES with different types of serially connected non-critical loads will first be introduced. Then, the various power delivery issues of the dc power systems, namely bus voltage variation, voltage droop, system fault, and harmonics, are briefly described. The operating limits of a DC-ES in a dc power grid is studied. It is demonstrated that the aforementioned issues can be mitigated using the proposed DC-ES technology. Experiment results are provided to verify the feasibility of the proposed technology.
Journal articleFesta A, Vinter RB, 2016,
This paper provides a decomposition technique for the purpose of simplifying the solution of certain zero-sum differential games. The games considered terminate when the state reaches a target, which can be expressed as the union of a collection of target subsets considered as ‘multiple targets’; the decomposition consists in replacing the original target by each of the target subsets. The value of the original game is then obtained as the lower envelope of the values of the collection of games, resulting from the decomposition, which can be much easier to solve than the original game. Criteria are given for the validity of the decomposition. The paper includes examples, illustrating the application of the technique to pursuit/evasion games and to flow control.
Journal articleKiziroglou ME, Elefsiniotis A, Kokorakis N, et al., 2016,
Aircraft sensors are typically cable powered, imposing a significant weight overhead. The exploitation of temperature variations during flight by a phase change material (PCM) based heat storage thermoelectric energy harvester, as an alternative power source in aeronautical applications, has recently been flight tested. In this work, the applicability of this technology to use cases with smaller and larger size specifications is studied by fabrication, testing and analysis of a scaled-down and a scaled-up prototype. Output energy of 4.1 J/g of PCM from a typical flight cycle is demonstrated for the scaled-down device, and 2.3 J/g of PCM for the scaled-up device. The higher energy density of the scaled down prototypes is attributed to the reduction in temperature inhomogeneity inside the PCM. The impact of super-cooling on performance is analyzed by employing a simulation model extended to include super-cooling effects. It is found that super-cooling may be beneficial for scaling down, in applications with slow temperature fluctuations.
Journal articleCarnevale D, Galeani S, Sassano M, et al., 2016,
We consider the problem of output regulation for LTI systems in the presence of unknown exosystems. The knowledge about the multi-frequency signals exosystem consists in the maximum number of frequencies and their maximal value. The control scheme relies on two main components: an estimation algorithm, to reconstruct the signal generated by the exosystem, and a controller, to enforce the output regulation property to the closed-loop system. To tackle the first task, we propose a hybrid observer for the estimation of the (possibly piece-wise continuous) number and values of the frequencies contained in the exogenous signal. The hybrid observer is particularly appealing for numerical implementations, and it is combined with a self-tuning algorithm of the free parameters (gains), thus improving its performance even in case of noisy measurements. Semi-global exponential convergence of the estimation error is provided. As far as the second task is concerned, a robust hybrid regulator is designed for practical rejection of the multi-frequency disturbance signal acting on the plant. The result is achieved by exploiting the frequencies estimated by the hybrid observer. The effectiveness of the proposed control scheme is shown by means of numerical simulations.
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