This study presents a chance-constrained scheduling model based on probabilistic and robust optimisation to handle the uncertainty of renewable energy generation and loads in microgrids. In order to generate appropriate scenarios, a large number of scenarios are generated by a Latin hypercube sampling Monte Carlo method and reduced by a fast forward selection algorithm. With the aggregated scenarios, a probabilistic scheduling model is established to obtain the expectation of schedules in different probability scenario. Aiming at taking full use of the generated scenarios, a robust optimisation is applied to the probabilistic model to consider the worst situations. The scheduling model proposed in this study combines the probabilistic and robust optimisation, in which the probabilistic one utilises the aggregated scenarios to introduce the probability characteristic of uncertainty and the robust one utilises the eliminated scenarios to consider the worst case of uncertainty. Finally, the proposed scheduling model is applied to a designed grid-connected microgrid, and the simulation results demonstrate the effectiveness of the proposed scheduling model.

The climatic factors have a strong influence on the polluted high voltage (HV) insulators performance and affect the reliability of networks. Such factors can lead to insulation incidents and outages in the electrical network. Based on monthly 12 weather factors observations (the average temperature, the precipitation, the humidity, the solar radiation, the vapour pressure, the snow depth, the dew point, the air pressure, the visibility, the total cloud cover, the wind direction and the wind speed) obtained during the period 2010–2015 (72 months) at Algerian central region, the authors propose an optimal relationship between the number of HV insulators incidents and the climatic factors. The autoregressive distributed lag (ARDL) method was employed to explore the actual and/or delayed (lagged) effect of each climatic factor on HV insulators performance. According to the results, the ARDL approach revealed the existence of strong relationship between these climatic variables and HV insulators incident. Also, the used statistical tests in this study allow us to say whether the proposed model is real and not simply due to chance. The proposed model has been validated by comparing the number of HV insulators incidents predicted for 10 months to real data recorded in Algerian AC transmission networks.

Voltage regulation (VR) and energy loss minimisation have always been major concerns for distribution network (DN) operators, thereby many conventional VR schemes are dedicatedly employed in existing DNs. In this study, optimal integration of different distributed energy resources (DERs) is investigated in coordination with existing VR scheme, i.e. on-load tap-changer. To show the superiority of the proposed DER integration model, optimal allocations of different DERs are determined with and without considering the coordinated effect of existing VR schemes for annual energy loss minimisation under different scenarios. To solve this complex optimisation problem, the improved genetic algorithm (GA) is adopted. A dynamic node priority list (DNPL) is suggested to further improve the performance of GA. To validate the proposed strategy and DNPL, the DER integration problem is solved for benchmark 33-bus and real-life 108-bus Indian radial distribution systems. The simulation results are found to be inspiring when compared with the existing optimisation techniques and DER integration models without considering VR schemes.

This study proposes a new adaptive and centralised under-voltage load shedding (UVLS) to prevent short-term voltage instability; it then examines the challenges related to the centralised UVLS. The proposed method uses a local measurement to estimate the amount of load shedding. It decreases the amount of load shedding by selecting the proper location of UVLS and shedding more powers within seconds after activating UVLS. Dynamic simulations are performed on the IEEE 118-bus test system, New England test system, and on Isfahan regional power grid (IRPG) as the case studies. Simulation results, when compared with the conventional multi-port network model and sensitivity-based methods, provide a considerable reduction in the active power shedding in addition to the number of the load shedding steps. Moreover, it indicates that the measurement bias errors have a considerable effect on UVLS. Finally, a method is introduced to overcome the effect of measurement errors.

The integration of renewable energy resources into the electrical distribution systems faces several stability challenges especially in the low inertia conditions. To address these issues, this study introduces a virtual synchronous machine (VSM) control strategy for the intertying power electronic converters in the autonomous AC/DC hybrid microgrids. It is shown that the VSM-based controller improves the system damping following the frequency disturbances and the AC/DC voltage variations. Moreover, a power management regulation topology is implemented in the active intertying converter to achieve an accurate bidirectional power flow under different loading conditions. A small-signal state-space model for the entire hybrid system is developed to assess the overall system performance. Time-domain simulation results under the PSCAD/EMTDC environment are also presented to investigate the effectiveness of the proposed techniques. The introduction of the VSM control for the intertying converters in the hybrid AC/DC microgrids provides a significant improvement in the dynamic performance and increases the robustness against external disturbances.

Audible noise is closely related to the frequency-domain characteristics of corona current. Based on the largest ultra-high-voltage (UHV) DC corona cage in the world and an actual UHV DC transmission line, a large amount of audible noise and corona current were measured simultaneously here. According to the statistical analysis of the experimental data, the correlation relationship between the frequency-domain characteristics of corona current and audible noise was obtained, and the causes of correlation rule were also analysed. Meanwhile, the correlation formula between the spectrum characteristics of corona current and A-weighted sound level of audible noise was discovered. The experimental results showed that the mathematical correlation formula discovered has good universality. The obtained results can provide theory and experimental foundation for deeply research on the essential connection between corona current and audible noise, which makes it possible to explore the causes of audible noise's change law from the perspective of ‘sound source’ and may provide a new method for audible noise estimation.

The DC distribution technology brings challenges and opportunities to the penetration of large-scale distributed renewable energy sources. In a medium-voltage DC (MVDC) distribution network, operation mode switch occurs owing to some large disturbances. To manage these disturbances and achieve seamless switch of the operation mode, a hierarchical control strategy with two layers, namely local and global layers, is proposed in this work. The novel local layer controller called P–U–I controller is designed to enhance the voltage stability, improve system controllability and suppress overcurrent under large disturbances. This controller only needs local information and does not rely on fast communication. Furthermore, as the scale of the DC distribution network expands continuously, the number of system operation mode increases in geometric progression. Therefore, a global layer controller based on breadth-first search algorithm is proposed. This controller can automatically identify the system topology and adjust the control mode of converters to optimise the system operating characteristics. Thus, this hierarchical control strategy can accurately control the system power in steady state, suppress overcurrent under large disturbances and suit large-scale DC distribution networks. Finally, a three-terminal MVDC distribution network simulation model established on power systems computer-aided design and RT-LAB validates the proposed strategy.

The variation of the electrical distance and the complexity of the electric network lead to the variations of feeder impedances between distributed generation units and load points. It is determined that conventional droop control has drawbacks in achieving accurate power sharing due to the effects of mismatched impedance. Therefore, this study proposes an enhanced proportional power sharing strategy based on adaptive virtual impedance in a low-voltage networked microgrid. The improved R–L type droop control can effectively prevent the coupling between real and reactive powers. Furthermore, an adaptive virtual impedance loop is introduced to counteract the feeder voltage drop. The method utilises real and reactive power mismatching which were fed into integral controllers, and then generates the virtual inductive and resistive components, respectively. This proposed strategy is able to enhance power sharing accuracy without requiring the knowledge of feeder impedance, and it is more adaptive to the complex impedance. The simulation experiments carried out under the environment of MATLAB/Simulink, and results verify the effectiveness of the proposed strategy.

A method to detect high impedance faults (HIFs) based on low-order harmonics is proposed. Short-time Fourier transform (STFT) is used to extract the main harmonic components of phase current, as magnitude and phase of the third harmonic and magnitude of the second and fifth harmonics, which are used to identify HIF occurrence. In addition, this study presents an analysis of the window length and type used in STFT and its suitability for the application. The method proposed is able to detect HIF occurrence at various points of the test system, as well as HIF happening on different soil surfaces. It is also capable of distinguishing HIF events from similar distribution system disturbances such as capacitor banks switching and feeder energising. Furthermore, it is effective when real HIF oscillography is applied, indicating its tolerance to typical noises present on real signals.

In this paper, an optimum recloser and fuse coordination scheme is proposed in the presence of multiple distributed generators (DGs) in a radial distribution network. The proposed approach formulates the recloser and fuse coordination problem as an optimisation problem and applies interior point method (IPM) to solve this optimisation problem for obtaining the optimum recloser and fuse settings. The proposed scheme gives a single set of the recloser and fuse settings which is robust enough to be able to coordinate the operations of the recloser and fuses properly without and in the presence of single/multiple DGs in the system. The effect of fault current limiter (FCL) on recloser–fuse coordination has also been studied. The proposed approach has been tested on IEEE 33-bus system for three different scenarios: (i) no DG in the system, (ii) a single DG in the system, and (iii) multiple DGs in the system. Further, the optimum results obtained by IPM have been validated by comparing them with those obtained by modular in-core non-linear optimisation system solver, sequential quadratic programming, particle swarm optimisation, harmony search, genetic algorithm, and differential evolution techniques. The test results prove the robustness and effectiveness of the presented scheme.

Current differential protection characterised by whole-line quick action has now become the main concept in the research of the wide-area relay protection algorithm. However, strict requirements on broadband and real-time capability are difficult to meet in actual projects. Aiming at reducing the system traffic while not compromising the protection performance, the study proposes an improved algorithm of wide-area current differential protection based on non-uniform quantisation. The algorithm is proposed based on the information theory and in view of the non-uniform distribution of information redundancy of current signal in different width intervals. Comparing with traditional current differential protections, the improved algorithm is able to retain the sensitivity and reliability of the protection while reducing at least 44% of the system traffic whenever internal or external faults occur. It facilitates the realisation of wide-area differential protection in practice. Through digital simulation and dynamic simulation, the study demonstrates that the algorithm can distinguish fault places correctly and features a relatively high degree of sensitivity.

Combined economic environmental dispatch problem (CEEDP) is one of the greatest challenges of the future smart grids. It aims at reducing the total cost during the power production process considering the growing environmental impact due to the emission of gaseous pollutants of fossil fuels. This study develops a robust distributed algorithm based on consensus protocols in multi-agent systems, to solve the smart grid CEEDP with a practical communication network consisting of a dynamic communication network, randomly communication failure, transmission delay and noise in communication channels. The proposed algorithm is fully distributed and cooperative in such a way that it eliminates the need for a central energy-management unit, or a leader. The performance of the fully decentralised consensus protocol was evaluated on the IEEE 30-bus and the IEEE 118-bus test system. A comparison with previous consensus algorithms proves the supremacy of the proposed approach in terms of its robustness under dynamic communication network with randomly link failure.

Expansion of cogeneration technologies, such as combined heat and power units, has boosted the growth of multi-carrier energy systems. A two-step method is presented to enable residential demand response (DR) programmes in the multi-carrier energy systems. In the first step, the energy management system at each home solves an optimisation problem to achieve the desired energy cost and demand schedule for the customer according to received price signals. In the second step, the system operator revises the demand scheduling by running another optimisation problem to minimise the total electrical losses, subject to the operational characteristics of electrical and natural gas systems. In order to persuade customers to participate in the DR programmes, it is guaranteed that the resulted cost of the second step is not more than the desired cost of the customer in the first step. Results of applying the proposed method, incorporating different penetration levels of customer participation in a time-of-use programme is studied in a test energy system. Simulation results verify the effectiveness of proposed method in minimising the total electrical losses, improving the operational characteristics of the energy system as well as providing customers' utilities.

The reactive power optimization (RPO) problem of power systems is a mixed-integer nonlinear programming problem, which can be computationally expensive. This paper applies the alternating direction method of multipliers (ADMM) to solve it. By duplicating discrete control variables with one copy allowed to vary continuously, the RPO model is formulated as a modified model in which the objective function and constraints have a separable structure with respect to the continuous and discrete variables, except for the coupling constraint. We then applied the ADMM to solve this model with the separable structure, and hence the original problem is converted into two optimization sub-problems, which are nonlinear programming (NLP) and mixed-integer quadratic programming (MIQP) problems, respectively. To improve the convergence of the algorithm, we proposed an extended ADMM by adding the upper and lower limits on state variables into the MIQP sub-problem based on the sensitivities obtained in the NLP sub-problem. We also established a mechanism to filter out inactive inequality constraints in the MIQP sub-problem to improve computational speed. Moreover, numerical results tested on IEEE test systems and a real 739-bus system have shown the correctness of the proposed method and its adaptability for power systems with different sizes and configurations..

In this work, a simple phase-locked loop – less control is presented for a single-stage solar photovoltaic (PV) – battery-grid-tied system. As compared to traditional solar PV systems, the system has reduced losses due to the absence of boost converter and a flexible power flow due to the inclusion of a storage source (battery). The synchronous reference frame theory is used to generate the pulses for switching the voltage-source converter (VSC), while maximum power is extracted from the solar PV array by using perturb and observe-based maximum power point tracking technique. The inherent feature of shunt active filtering by the VSC has also been incorporated in this system. Test results for the system operation under fixed power and variable power mode are studied on a prototype developed in the laboratory. During fixed power mode, a fixed amount of power is fed to the grid, whereas in variable power mode the power fed to the grid varies. Test results obtained are in accordance with the IEEE-519 standard. This work is a basis for the upcoming power market, where solar PV consumers can manage the generated electricity and maximise their profit by selling the power to the grid judiciously.

Accurate renewable resource and load forecasting plays a key role in the progress of power grid planning schemes. In this study, a hybrid short-term forecasting method based K-means clustering and variational mode decomposition (VMD) technique is proposed to deal with the problem of forecasting accuracy. K-means clustering is a means of data mining approach and used for classifying data into several clusters. A cluster selection method is adopted to extract similar features from historical days. To better analyse the time series of historical data, VMD decomposes time series data into an ensemble of components with different frequencies. Self-adaptive evolutionary extreme learning machine as a novel and fast regression tool is trained and used for predicting each component. Eventually, the forecasting result generated by reconstructing all the predicted components values. The performance of the proposed hybrid forecasting model is evaluated by using real data from National Renewable Energy Laboratory. The simulation results show that it can obtain better forecasting accuracy than some previously reported methods.

Dissolved gas analysis (DGA) of oil is one of the major diagnostic tools that are used to detect the incipient faults of oil-immersed transformers through the correlation between the dissolved gases content in transformer oil and a particular malfunction. There are several techniques of DGA to interpret the incipient transformer faults such as Doernenburg ratio, Rogers's ratio, IEC ratio and Duval triangle techniques. This study presents a new proposed technique for transformer fault diagnosis using a new three gas ratios concentration of DGA to overcome the conflict that takes place in the traditional interpretation techniques. The accuracy of the proposed diagnosis technique is verified using practical DGA data obtained from 688 samples that have been collected from different transformers of different rating and different lifespan reported by Egyptian Electricity Holding Company, IEC TC10 and related databases surveyed from actual incident cases. The comparison with traditional DGA techniques showed that the proposed DGA ratio technique has good diagnosis accuracy. It is found that the suggested three ratios technique method has 99.86% accuracy as compared with 85.67% for Duval, 75.08% for Doernenburg, 47.34% for IEC and 39% for Rogers when testing 688 cases.

A doubly fed-induction generator (DFIG) has a significant advantage of fast dynamic frequency regulation. However, conventional virtual inertia control (VIC) design only takes frequency into account and may face secondary frequency drop during rotor speed restoration. To solve these problems, the novel decentralised control strategy is proposed for a hybrid micro-grid. Firstly, the load disturbance is considered to enhance control system accuracy, which is estimated by using the state observer based on the dynamic mathematical model. Then, the sliding mode VIC controller is designed by taking advantage of the observed disturbance value and frequency deviation for DFIG output dynamic adjustment according to the conventional frequency control (VIC) model. Furthermore, during DFIG rotor speed restoration, pitch angle control based on the mathematical model is applied for reserve capacity, and a new triggering condition according to the conventional condition is designed, so as to release reserve power at an appropriate time. Finally, the proposed strategy is tested in a typically isolated micro-grid with diesel and DFIG through a simulation and experiment.

A bang-bang phase angle controller (BPAC) was proposed in this study for the primary frequency control of doubly-fed induction generator-based wind turbines (DFIG-WTs). Dynamics of the internal voltage of a synchronous generator (SG) and that of a DFIG-WT were investigated in frequency deviation events. A BPAC was designed to regulate the phase angle obtained with a phase-locked loop directly, which enables the rapid active power control of the DFIG-WT. The BPAC signal is fed into the active power regulation loop of the pitch angle controller, which is expected to help rotor speed recovery and prevent secondary frequency drop. The small-signal analysis was carried out for the closed-loop system, composed of the DFIG-WT and the external SG-based power system, to verify the stability of the overall system. Simulation studies were undertaken on a wind power penetrated multi-machine power system, through which the primary frequency control performance of the BPAC was verified.

Probabilistic production simulation is an important tool used in conventional power systems to calculate generated energy and evaluate reliability. With the continuous expansion of the renewable energy plants, new characteristics such as intermittency and volatility of power are widely discussed, and it is necessary to study the application of probabilistic production simulation for renewable energy power generation. This study proposes a probabilistic production simulation method based on sequence operation theory (SOT) to simulate the operation of a wind/photovoltaic/energy storage power system. Both the uncertainty of renewable resources and the outage of wind turbines are considered in this study. Considering the complementary property of the renewable energy, pattern clustering is used to analyse the meteorological conditions and to assist in the probabilistic production simulation. Moreover, the output model of the energy storage device is developed using the Monte Carlo method and controlled using a smoothing strategy of the energy storage device. Ultimately, the simulation example shows the feasibility and the higher efficiency of the algorithm compared with Monte Carlo method and a production simulation method based on equivalent energy function.

A coordinated look-ahead reactive power optimisation method is proposed to minimise the required number of operating control devices for a time horizon of 24 h. The aim is to determine, via solving a mixed integer non-linear programming (MINLP) problem, optimum value settings of transformer taps, capacitor banks and reactive power output of distributed generators (DGs) based on the day-ahead load demand and active power output of DGs satisfying the engineering and operational constraints. The proposed method employs a three-stage method: assessment stage, time-period-partitioning stage, and coordinated reactive power optimisation stage. The first stage assesses the hourly voltage profile and available delivery capability margin of the system, while the time-period-partitioning stage uses clustering algorithm based on power-flow solution to partition time periods into coherent time durations. The MINLP problem is solved in the proposed coordinated optimisation stage. A modified IEEE13 case and IEEE123 case are used to verify the effectiveness of the proposed three-stage method.

This study presents simulation and experimental verification of solid-state switch overvoltage protection. The switching device is composed of two antiparallel connected forced commutated semiconductor valves and in analysed case has been used as a low-voltage solid-state bypass switch of UPS system. The aim is to identify the value of overvoltage induced on switch terminals and surge energy during an instance of current turn-off in line with inductive impedance. Afterwards determine optimal solution that protects such a power electronics device. As a protection solution configurations and types of surge arresters have been examined. All simulations have been conducted using PSCAD software package on the model especially prepared for this purpose.

This work studies the power quality impacts of electric transportation charging including electric buses (EBs) and electric vehicles (EVs) on distribution systems. The stochastic harmonic models for EBs and EVs and the stochastic models for their usage scenarios are built utilising the measured charging data. Based on these models, a power quality analysis method considering the charging uncertainties of EBs and EVs is proposed in this study and then their impacts on distribution systems are investigated. The Monte Carlo simulation is integrated into the proposed method to study the under voltage, overcurrent, voltage unbalance, transformer overload, harmonic current, voltage etc. caused by the stochastic charging of EBs and EVs. An actual distribution system acquired from Taiwan Power Company is taken as an example to validate the practicability of the proposed method. The test results of the proposed stochastic model are compared with those of the conservative model.

A probabilistic load flow (PLF) method based on the sparse polynomial chaos expansion (PCE) is presented here. Previous studies have shown that the generalised polynomial chaos expansion (gPCE) is promising for estimating the probability statistics and distributions of load flow outputs. However, it suffers the problem of curse-of-dimensionality in high-dimensional applications. Here, the compressive sensing technique is applied into the gPCE-based scheme, from which the sparse PCE is built as the surrogate model to perform the PLF in an accurate and efficient manner. The dependence among random input variables is also taken into consideration by making use of the Copula theory. Consequently, the proposed method is able to handle the correlated uncertainties of high-dimensionality and alleviate the computational effort as of popular methods. Finally, the feasibility and the effectiveness of the proposed method are validated by the case studies of two standard test systems.

This study presents a new powerful method for probability density function (PDF) estimation of the probabilistic power flow problem results. The proposed method works on a prior density function to improve its potential PDF estimation. The generalised cross-entropy method is used to refine the initial dataset. Most of the existing methods for PDF estimation do not provide a sparse result and rely on approximation and simplification. Compared to the other conventional methods, the proposed method has some remarkable features in density estimation. Low computational burden or high-speed response and high-precision results are the most prominent features of the proposed method. The method can deal with a correlated problem which is considered the correlation between uncertain parameters. It is tested on IEEE 14-bus and IEEE 118-bus systems, considering the non-stationary loads, some renewable energy resources and their correlations. The results of the proposed method are compared against more accurate results obtained from Monte Carlo simulation and some other conventional methods in density estimation. Comparisons confirm the high accuracy level and speed of the proposed density estimator against the other methods.

Installing magnetic shunts on the surfaces of tank and clamps of power transformer is able to reduce the eddy current losses and decrease the temperature rise effectively. Here, the three-dimensional transient finite element method and the magnetic thermal coupling method are used to calculate and analyse the effect of magnetic shunts with different structures. The non-linear magnetic property of both the transformer structural parts and the magnetic shunts are considered in the analysis. The lamination structure of the shunt is also considered, which is simulated by a thick plate with anisotropic magnetic property. A 334 MVA single-phase transformer is taken as an example in the analysis. The optimisation shunt structure is obtained. By comparing the L-shaped, the inverted L-shaped, and the U-shaped magnetic shunts for the clamps, it is proved that the U-shaped shunt is more effective in reducing the eddy current loss and temperature rise.

Temperature has obvious effects on the tested results of frequency domain dielectric spectrum (FDS) method. To estimate the moisture content of field oil-immersed bushing, the effects of temperature must be considered and eliminated. In this study, the frequency-domain dielectric spectrum of 72.5 kV bushing samples with different moisture contents was tested at different temperatures. The law that temperature affects the results of FDS tests was analysed and concluded. Afterwards, a method to transfer the tested results to the reference temperature (15°C) was proposed. Finally, an estimation method of moisture content for field oil-immersed bushing was proposed and testified. The following conclusions can be concluded: (i) the effects of temperature on the FDS results may be divided into two aspects, i.e. one is along the horizontal axis and the other is along the vertical axis and (ii) based on the proposed method, the temperature effects on the results of FDS tests can be eliminated, and the moisture content of field oil-immersed bushing can be estimated effectively and feasibly.

Advances in metering and two-way communication technologies foster the studies of Home Energy Management System (HEMS). This study proposes a new HEMS, which optimally schedules the distributed residential energy resources (DRERs) in a smart home environment with varying electricity tariff and high solar penetrations. The uncertainties of solar power output are captured by using Monte Carlo sampling technique to generate multiple solar output scenarios based on the probabilistic solar radiation model. The homeowner's rigid and elastic restrictions on the operations of the automatically controlled household appliances are modelled. Based on this, an optimal DRER scheduling model is proposed to minimise the home operation cost while taking into account the homeowner's requirements. A new heuristic optimisation algorithm recently proposed by the authors, i.e. natural aggregation algorithm, is used to solve the proposed model. Simulations based on real Australian solar data are conducted to validate the proposed method.

Voltage sag is the most common and severe power quality problem in the recent times due to its detrimental effects on modern sensitive equipment. Generally, direct-on-line starting of the three-phase induction motor (IM) and various kinds of short circuit fault are directly responsible for this event. This study investigates the impacts of starting and stopping of two three-phase IMs on the load voltage profile. To be more critical, two three-phase short circuit faults and one unsymmetrical fault are also simulated in the same network at different instants of time. A simple control algorithm of a real power optimised dynamic voltage restorer (DVR) with a reduced power factor strategy is presented to protect the sensitive load from these types of detrimental events. A novel fuzzy-proportional–integral based self-tuned control methodology is implemented in the proposed work to compensate the loss in the DVR circuit as well as to regulate the load voltage and the direct current link voltage. The results show the effectiveness of the adopted control scheme in DVR application to mitigate the voltage sag.

This study presents an evaluation of the application of the controlled switching device for reducing the level of inrush current and transient voltage dip (TVD) during energisation of an unloaded power transformer using circuit breaker having pre-insertion resistor (PIR-CB). This evaluation is achieved by analysing the effect of statistical scatter of mechanical operating time (MOT), tolerance of auxiliary contact used as a feedback for a controlled switching device, and variation in mechanical insertion time (MIT) and electrical insertion time (EIT) of the PIR-CB. The modelling of a complete system consisting of a power source, PIR-CB, and two types of power transformers (electrically and magnetically coupled) is carried out using power systems computer-aided design/electro-magnetic transient design and control (PSCAD/EMTDC) software package. The results obtained from the simulations are compared with the results obtained during energisation of similar transformers in the real field. Finally, the performance of PIR-CB-based proposed methodology has been compared with the conventional methodology (non-PIR-CB) with reference to mitigation of inrush and TVD. The close match between field results and that obtained with simulation for both electrically and magnetically coupled transformers authenticate the proposed methodology.

Frequency oscillations with very low oscillation frequencies were observed in several power systems. In frequency oscillations, the speeds of all generators change coherently and the frequencies in the system oscillate together. An equivalent model, which only preserves the governors, turbines, and an equivalent generator, is subsequently built. In the model, the inter-machine rotor-angle oscillation modes are eliminated and only the frequency oscillation mode is preserved. First, a simplified relationship between the total generator output power and frequency is adopted to obtain an original reduced-order model. With the model, an initial estimation of the eigenvalue corresponding to the frequency oscillation mode is obtained. The initial estimation is used to obtain a more accurate relationship between the total output power and frequency, and then an improved reduced-order model is built. With the model, a more accurate eigenvalue estimation can be obtained. Owing to the low order of the proposed model, the computational burden is greatly reduced compared to the direct analysis with fully modelled systems. The test results verify the validity of the proposed method.