*Article* **Hybrid-Model-Based Digital Twin of the Drivetrain of a Wind Turbine and Its Application for Failure Synthetic Data Generation**

**Ainhoa Pujana 1,\*, Miguel Esteras 1, Eugenio Perea 1, Erik Maqueda <sup>1</sup> and Philippe Calvez <sup>2</sup>**


**\*** Correspondence: ainhoa.pujana@tecnalia.com

**Abstract:** Computer modelling and digitalization are integral to the wind energy sector since they provide tools with which to improve the design and performance of wind turbines, and thus reduce both capital and operational costs. The massive sensor rollout and increase in big data processing capacity over the last decade has made data collection and analysis more efficient, allowing for the development and use of digital twins. This paper presents a methodology for developing a hybrid-model-based digital twin (DT) of a power conversion system of wind turbines. This DT allows knowledge to be acquired from real operation data while preserving physical design relationships, can generate synthetic data from events that never happened, and helps in the detection and classification of different failure conditions. Starting from an initial physics-based model of a wind turbine drivetrain, which is trained with real data, the proposed methodology has two major innovative outcomes. The first innovation aspect is the application of generative stochastic models coupled with a hybrid-model-based digital twin (DT) for the creation of synthetic failure data based on real anomalies observed in SCADA data. The second innovation aspect is the classification of failures based on machine learning techniques, that allows anomaly conditions to be identified in the operation of the wind turbine. Firstly, technique and methodology were contrasted and validated with operation data of a real wind farm owned by Engie, including labelled failure conditions. Although the selected use case technology is based on a double-fed induction generator (DFIG) and its corresponding partial-scale power converter, the methodology could be applied to other wind conversion technologies.

**Keywords:** wind turbine; digital twin; hybrid model; failure diagnosis; synthetic data generation; predictive maintenance

#### **1. Introduction**

In modern times, wind energy conversion is one of the most promising and reliable energy technologies. Europe already has 220 GW of wind capacity installed and there are plans to install an additional power of 105 GW over the next five years [1]. Actors involved in this energy source are continuously researching this technology with the aim of achieving the best levelized cost of energy (LCOE). According to WindEurope, operation and maintenance (O&M) expenses account for 25–35% of LCOE of wind turbines [2], where corrective maintenance is responsible for 30–60% of O&M costs [3]. The current potential of digitalization and artificial intelligence (AI) can greatly contribute to the increase in the energy production of wind farms, reducing unplanned interruptions, optimizing O&M, and extending the lifetime of the components.

Wind turbines systems can be classified depending on the type of generator, gearbox and power converter used. A double-fed induction generator (DFIG) with a multiple stage gearbox and a partial scale converter is a widely used technology [4]. In the DFIG topology [5], there is a direct connection between the stator windings and the constant

**Citation:** Pujana, A.; Esteras, M.; Perea, E.; Maqueda, E.; Calvez, P. Hybrid-Model-Based Digital Twin of the Drivetrain of a Wind Turbine and Its Application for Failure Synthetic Data Generation. *Energies* **2023**, *16*, 861. https://doi.org/10.3390/ en16020861

Academic Editor: Francesco Castellani

Received: 17 November 2022 Revised: 30 December 2022 Accepted: 1 January 2023 Published: 12 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

frequency grid while the rotor winding connection to the grid is made through a pulse width modulation (PWM) power converter, using a set of slip rings. The power converters can control the rotor circuit current, frequency, and phase angle shifts [6]. This kind of induction generator can operate in a range of ±30% of synchronous speed, achieving a high energy yield, a power fluctuation reduction and the capability of controlling reactive power. A drawback of the DFIG is the inevitable need for slip rings.

A wind turbine is also equipped with a control system, which is responsible for assuring the correct operation of the wind turbine along its entire power curve and keeping the wind turbine within its normal operating range. Wind turbines contain electrical, mechanical, hydraulic, or pneumatic systems, and require sensors to monitor the variables that determine the required control action. The most common variables sensed in a control system are wind speed, rotor speed, active and reactive power, voltage, and the frequency of the wind turbine's connection point. Moreover, the control system is responsible for stopping the wind turbine if necessary. One control strategy is the pitch angle control [7], which is a good option for variable-speed operations in wind turbines generating more than 1 MW. Using this control, the blades can be correctly oriented with respect to the wind direction in order to avoid extremal values (too high or too low) of the power output. The pitch system is based on a hydraulic system, which requires a computer system or an electronically controlled electric motor.

There are several studies that analyse the critical failure modes of the wind turbine drivetrain system, specifically the electric generator and power conversion system [8–10]. While identifying the sources of failure in the electric generator [11], the typologies of failures can be of different kind. Thermal failures can occur due to the effect that currents and overcurrents circulating through the windings have on the insulation and considering that a maximum temperature is withstood depending on the type of insulation and operating conditions. Electrical failures can also occur due to the peaks of voltage that can be applied to the conductor under normal operating conditions and in anomalous situations, such as surges coming from the converter. Environmental failures can be caused by environmental conditions that could degrade insulating material or create corrosion phenomena. Mechanical failures are mainly caused by vibrations. Finally, thermo-mechanical failures are caused by cyclic operating conditions with sudden or continuous variations in temperature, which have different effects depending on the cable material and its accessories (insulation, screens, etc.). The electric generator and the power converter have a greater impact on the reliability, failure rate, and unavailability of the wind turbine. Their failure rate is 15% per year for the electric generator and 6.8% for power converters of offshore wind farms [12,13]. These components are equipped with sensors (temperature, vibrations, electric parameters and others) and connected to the wind turbine supervisory control and data acquisition (SCADA) and condition-based monitoring (CBM) systems. Thus, a long historical real operation dataset exists for each turbine of a wind farm. Sometimes, this dataset includes recorded anomalies or failure in the operation of the turbine.

Data-driven models extract knowledge from real measurements that apply AI (artificial intelligence) techniques, which analyse large amounts of data to identify meaningful patterns in them. In the field of wind energy generation, there are several approaches for this type of model. For instance, the spectral analysis of current signals has been used for health monitoring of stator and rotor windings, as well as the main bearing of wind turbines [14]. In [15], a data-driven model is directly constructed with the objective of detecting and isolating sensor and actuator failures in wind turbines, while the study of [16] develops a hierarchical bank of negative selection algorithms (NSAs) to detect and isolate common failures in wind turbines. The study of [17] uses a data-driven failure diagnosis and isolation (FDI) method for wind turbines. It consists of the implementation of long short-term memory (LSTM) networks for residual generators. The decision-making process is made by applying a random forest algorithm. These FDI methods are designed using experimental and historical data generated both under normal and failure conditions; therefore, the availability of well-developed databases that include labelled anomaly/failure data is

mandatory. The accuracy of data-driven methods is generally poor for cases not included in a training dataset. In addition, black box models (e.g., deep learning models) show a low explainability, making it difficult for domain experts to interpret results and gain the required trust to make decisions based on the output of the models.

As a solution to this main drawback of data-driven models, DTs that use physicsbased models are developed to make the DT self-explanatory. The term "digital twin" can be defined as "a virtual representation of a real-life system or asset with the same behaviour". It allows system states to be calculated using integrated models and data, aiding the decision-making process over its life cycle from design to decommissioning. The concept of DT was first described in David Gelernter's 1991 book *Mirror Worlds* [18], and the term "digital twin" was first mentioned in a roadmap report developed by John Vickers (NASA) in 2010. The DT concept consists of two distinct parts: (1) the physics-based model representing the asset and (2) the connection of the model with the real asset. This connection refers to the information transferred (automatically or manually) from the asset to the DT and the information that could be transferred from the DT to the asset and the operator. In this way, a DT can accurately estimate an asset's condition.

A DT is based on mathematic models that represent physical phenomena, making it possible to understand the behaviour of the real asset in each moment. In addition, using this physics-based model, it is possible to create synthetic data for events that have never happened before, acquiring knowledge of the behaviour in some conditions that in other cases would not be possible. Data-driven models can identify and prevent events that were measured in the past. However, the training process of the data-driven algorithms, either non supervised or supervised, always relies on historical data. DTs, on the contrary, provide two new information sources: firstly, physics-based models can allow us to understand their real behaviour, and secondly, physical simulation enables the generation of synthetic data for potential new scenarios, such as potential anomalies or failure conditions. Moreover, hybrid models, considered to be a combination of physics-based models and data analytics, provide a powerful tool for diagnosis and prognosis [19]. Hybrid models developed with this purpose are a good basis for DT creation.

The main advantage of a DT design for a specific industrial setting is the potential to simulate realistic scenarios that are difficult or costly to create in the real system. These scenarios might be used for the prescriptive analysis of new operating conditions, or for testing extreme conditions and responses to anomalies or failures. The main challenge is to develop a simulation method that can be parametrized to output scenarios that differ from normal operation and, in some cases, to simulate conditions that have never been seen before in the real system. The authors of [20] describe four main approaches for the generation of simulated scenarios based on: (1) a simplified physical model; (2) a more complex DT design to model the specific properties of the real scenario; (3) a parametrized statistical generative model built upon prior knowledge of the relationships between variables; and (4) generative models trained with existing real data distribution.

The methodology proposed in this paper brings together approaches 2 and 4 to develop a hybrid digital twin that combines physics-based models and data-driven models to match a specific operation context, both in normal and extreme or failure conditions. In addition, the DT preserves the constrains, significance and explainability of a physical model, overcoming some of the main limitations of a purely statistical generative model (i.e., generative adversarial networks). The physics-based model for the drivetrain of a wind turbine is developed using MATLAB Simulink R2020b.

The paper is organized as follows: Section 1 describes the developed technical approaches and the literature review related to such technical approaches, as well the problems of using data-driven approaches in comparison with hybrid models. Section 2 explains the proposed methodology for developing a hybrid-model-based digital twin and the advantages of combining both physics-based and data-driven models. Moreover, this section describes the principles of synthetic data generation and how such principles can be applied to failure data generation. In Section 3, this methodology is concretely applied

to a use case: the drivetrain of a 1.5 MW wind turbine with DFIG technology. Section 4 contains the conclusions and perspectives of future research.

#### **2. Methodology for a Hybrid Model Creation, Synthetic Failure Data Generation and Failure Classification Applied to a Digital Twin**

DT development involves several technical tasks combining domain-specific knowledge and data analytics skills. First, the equipment or system deterministic model in normality conditions (so-called normality model) must be generated (e.g., by simulation model). This process includes the representative modelling of underlying physical phenomena and the rigorous selection of design parameters. Then, the constructed model must be validated using real data in non-failure conditions and optimizing certain model parameters values to increase the model accuracy and representativeness against the real equipment behaviour.

In addition, a DT conceived for failure conditions diagnosis includes a suite of physicsbased models able to simulate different anomaly or failure scenarios. These failure models might be used for a cause–effect analysis and to establish condition indicators (CI) and they constitute an excellent basis for real failure conditions synthetic data generation [21]. Finally, machine learning (ML) classification techniques (supervised or non-supervised) might be applied for the diagnosis or early detection of failures. The implementation of all these models and algorithms in a digital platform and their online use constitute a complete DT for anomaly/failure diagnosis.

This chapter describes and analyses the methodology for the development and use of an equipment or system DT based on hybrid models for failure classification, making use of a normality hybrid model and a synthetic data generation process. Figure 1 summarizes the whole methodology, and each key component is explained in the following chapters.

**Figure 1.** Methodology illustration for the creation of a hybrid-model-based DT.

#### *2.1. Normality Hybrid Model*

The normality hybrid model of the DT is composed of a physics-based model trained with real operation SCADA data in normality conditions.

The paper considers the drivetrain of a wind turbine with DFIG technology as a reference use case in which the proposed DT development methodology is illustrated and applied. Figure 2 shows how the physics-based model is divided in two modules that could be used either coupled together or separately, depending on the available operational data. The first module represents the conversion from kinetic energy from the wind to mechanical power, taking the real values of the wind speed measured at the turbine and the pitch angle of the blades as inputs. The second module represents the electro-mechanical conversion. It takes the mechanical torque in the shaft of the DFIG as the input and the generated electric power and its related signals, such as phase currents and voltages or electromagnetic torque, are the outputs. Moreover, this second module includes a power converter and control system that enables the optimal operation of the drivetrain.

**Figure 2.** Physics-based model of the power conversion drivetrain of a wind turbine.

The physics-based model is constructed considering the system design parameters. Depending on the nature of the equipment it may be difficult to obtain the complete set of design parameters. In this case, estimations are required, which may impact model performance. Finally, the physics-based model is trained using real operation SCADA data (Figure 3). Training consists of optimizing the values of certain independent design parameters whose exact values are estimated between given realistic intervals.

**Figure 3.** Training of the physics-based model and obtention of the normality hybrid model.

The objective function of the training process is the minimization of "residue" defined as the difference between the physics-based model output (prediction) and the SCADA real operation data (e.g., output power) for the given real inputs (e.g., wind speed or torque). The resulting calibrated physical model is known as the normality hybrid model.

#### *2.2. Failure Hybrid Model*

Once the normality hybrid model is constructed, it can be extended or adapted to include anomaly or failure situations. This new model is called a failure model. Following the same process used in the normality hybrid model, this model is trained using the operation real SCADA data. Similarly, calibration consists of optimizing the values of certain independent design parameters that represent failure, whose exact values are estimated between given realistic intervals.

This resulting new model is also trained with historical and actual operational data of both normal and failure operation. This is achieved using real failure operation data inputs, which are fed to the failure models. In other words, when the normality hybrid model is adapted to represent a failure and trained with failure data (data representing failure operation), the normality hybrid model becomes a failure hybrid model. Feeding the failure models with failure data enables the values of the failure model parameters that define the failure models to be calibrated. The selected values of these failure model parameters are obtained by minimizing the difference between the prediction obtained by the failure model using failure operation data inputs and their corresponding well-known real operation data failure outputs. As a result, the so-called failure hybrid model of the power conversion system (drivetrain) of a wind turbine is obtained, which considers both data of the drivetrain in normal operation and in failure operation.

In this case, the overheating of the DFIG stator winding is studied. For this scenario, a thermal model is added to the normality hybrid model (Figure 4).

**Figure 4.** Failure hybrid model with a specific thermal failure model.

This thermal model takes as input the real values of the nacelle temperature and the stator phase currents. These values of these stator currents can be estimated by the normality hybrid model or any other value that can be useful for testing the thermal behaviour of DFIG stator windings. The obtained predicted output corresponds to the temperature of the DFIG stator winding.

#### *2.3. Failure Synthetic Data Generation*

The methodology analysed in the article has a fundamental contribution in the generation of synthetic data. The generation of synthetic data is a key point because it allows immediate availability of operation data (either normality or failure data), that are difficult to obtain from simple observation of the reality. In addition, the training of classification models for failure prognosis is much enriching using a broad and balanced dataset that represents a variability of behaviour.

Ref. [22] proposes GANs for the generation of synthetic data for wind turbine failure diagnosis research. This article proposes a method to generate synthetic data using the hybrid model and a statistical process. The statistical process characterizes the probability distributions of the occurrence of normal and failure operating scenarios.

The generation of synthetic scenarios in a DT is often deterministic; therefore, the given input data (i.e., wind speed, nacelle temperature and blade pitch angle) always calculate the same output data (i.e., active power, winding temperature, etc.). This process does not consider the variance present in the real data due to factors not modelled by the DT. Hence, the DT does not have the ability to interpolate within the space of the training data and cannot generate truly new scenarios, nor can it include the full extent of the variability observed in the data. In the case of the generation of normal condition scenarios, this determinism is compensated by the amount of training data in such conditions. It is reasonable to assume that these data include a comprehensive range of conditions that represent the entire feature space.

However, this might not be the case for the generation of failure conditions. Although the failure hybrid model has been calibrated to simulate the instances belonging to this type of conditions present in the training SCADA data, this does not guarantee that these instances are a representation of the entire anomalous feature space. In fact, the frequency of anomalous conditions and failures is relatively low in SCADA data, and often these instances are not annotated (labelled). Hence, relaying merely on a deterministic model to generate synthetic failure scenarios would provide a narrow data sample constrain to patterns already seen before.

To resolve this limitation, the DT can incorporate stochastic failure models for the generation of failure scenarios. Each of these models can generate an unlimited number of synthetic failure scenarios for a particular failure type based on real observations in SCADA data.

The corresponding models are trained to approximate the distributions of the variables that define a failure. In addition, some failures cannot be considered instantaneous, but as a pattern in time that leads to a malfunction, a safety stop or a break. This is especially important if synthetic generated failures are to be used to train models that can produce early warnings before a failure is likely to occur.

Both the join probability distribution of the operating variables prior to and during a failure and their physical constrains are initially defined by domain knowledge and can then be updated with observations from real SCADA data. The generation of new failure scenarios is based on random sampling of these probability distribution. Hence, the synthetic scenarios generated by the model are based on real SCADA observations but are not identical to any of those. The process for the synthetic failure data generation of Figure 1 is detailed in Figure 5. It consists of two steps: an observation step and a synthetic data generation step. The observation step aims to identify the probability density function (PDF) that characterizes the failure scenario occurrence. For this, SCADA data are filtered to identify scenarios that correspond to a failure type *fk*, where k is part of a set of failures K modelled by the DT, such that *k* ∈ *K*. A failure scenario is defined by a set of fixed physical constrains defined by domain knowledge and a set of parameters (condition indicators) to be tuned in function of the observed features in failure scenarios from SCADA data.

**Figure 5.** Observation process for failures.

The PDFs of the parameters are learnt from the observed instances in the SCADA data. These instances might be exclusively sourced from a single turbine or, in case of an insufficient number, they can be sourced from different turbines that share some design and operations characteristics. The decision to include instances from more than one turbine should be made on the basis of turbine similarity and the variability of failure parameters, which depends on operation and design characteristics. The distribution of most parameters might be approximated by a normal PDF with the required precision. However, other distributions might need to be considered for certain parameters. In the case of having access to SCADA data with several instances of a given failure for more than one turbine, a hierarchical parameter modelling might provide a better balance between accuracy and generalization. The learnt PDFs of the parameters are used to update the prior parameter distributions of the corresponding failure model. The data generation process step consists of generating data sets for normality and failure scenarios. As shown in Figure 6, the normality scenario data sets are generated either by running the normality hybrid model or selecting those SCADA data labelled as normal data.

**Figure 6.** DT generative failure models.

The failure scenario data sets are stochastically generated following the observed and identified PDF, then running and obtaining the results from the failure hybrid model.

#### *2.4. Potential Application of the Hybrid Models Conforming the Digital Twin*

The development of data-driven algorithms for diagnosing normality or failure conditions is a complex task that involves: (i) defining the condition indicators (CIs), (ii) labelling normality and failure operation data, (iii) conceptualization of the classification model, (iv) validation of the model (e.g., number of false positives and negatives), and (v) evaluation of the generalization capacity of the model analysing whether it is representative for a set of machines. The DT can add value to this endeavour by providing additional synthetic data to strengthen the dataset.

Figure 7 shows a proposed schema of a supervised classifier training process for failure diagnosis where the explained models in the previous sections are leveraged. The classifier is trained with a labelled dataset composed of real SCADA data, augmented with synthetic data generated via the process described in the previous section.

**Figure 7.** Supervised classifier training scheme.

In addition, the normality hybrid model is used as a baseline to create new CIs that may improve the accuracy of the classifier. These CIs are calculated by comparing real operation SCADA data with respect to synthetic failure data and/or normality data generated by the normality hybrid model.

Finally, Figure 8 shows the execution phase, where CIs are created by comparing real SCADA data with the data simulated by the normality hybrid model. When the values of these CIs meet certain criteria detected by the classifier, an early alarm is generated.

**Figure 8.** Execution phase of the developed classifier for anomaly diagnosis.

#### **3. Results of Application of the Methodology to a Use Case: 1.5 MW DFIG Wind Turbine**

The methodology described in previous section was applied and validated with real SCADA data from a wind turbine in operation owned by Engie. The drivetrain of this wind turbine comprises a 1.5 MW DFIG and its corresponding back-to-back power converter.

Three years of real operational data were organized and preprocessed before use. During the data exploration and pre-processing of SCADA data, relationships between physic parameters were analysed, in order to detect possible outliers, which were removed.

Once the initial data analysis was carried out, the physical model of the power conversion was developed in Simulink-Matlab R2020b (Figure 9). Information on the design parameters of both the generator and power converter was used as a basis for constructing the model. However, some other values were calculated or estimated due to the lack of information. Wind speed and pitch angle are the input parameters needed to operate the model. The result is the generated electric power, currents, and voltages, among others.

**Figure 9.** Wind turbine drivetrain physics-based model representation in Matlab-Simulink.

The DFIG block implements a three-phase wound rotor asynchronous machine, operating in the generator mode. It uses a fourth-order state-space model to represent the

electrical part of the machine, whereas the mechanical part is represented by a second-order system. As can be seen in the equations contained in Table 1, all the electrical parameters are referred to in the stator. All the rotor and stator parameters are expressed in the arbitrary two-axis reference dq frame.

**Table 1.** Equivalent circuits and equations involved in a DFIG conversion.

#### **Electrical System**


The parameters involved in the resolution of DFIG conversion equations are those indicated in Table 2.



#### *3.1. Normality Hybrid Model of the Use Case*

The initial parameters of the physics-based model are an assumption of the true parameters controlling the operation of a given turbine. Nevertheless, the true value of these parameters can be estimated using an optimization algorithm. The algorithm aims to find the combination of parameter values that minimize the difference between the output of the physics-based model and the measured SCADA data. In this case, the parameters are tuned (or calibrated) using a surrogated optimization algorithm (surrogateopt) in Matlab [23]. This optimization algorithm is a global solver specially indicated for cases where the objective function is computationally expensive. The algorithm searches for a global minimum of a cost function min*<sup>x</sup> <sup>f</sup>*(*x*) with multivariate input variable *<sup>x</sup>* subject to linear and non-linear constrains, and some finite bounds. The resulting objective function can be non-convex and non-smooth. The algorithm starts by learning a surrogate model of the function considered as objective, using the interpolation of radial basis function through random evaluations of the objective function within the given bounds. In the next phase, a merit function is minimized by approximating the minimization of the objective function. This merit function *fm* is based on a weighted combination of the evaluation of the surrogate model calculated in the previous phase, and the distance between the points sampled from the objective function.

$$f\_{\mathfrak{m}}(\mathbf{x}) = w\mathbf{S}(\mathbf{x}) + (1 - w)D(\mathbf{x}) \tag{14}$$

$$S(\mathbf{x}) = \frac{\mathbf{s}(\mathbf{x}) - \mathbf{s}\_{\min}}{\mathbf{s}\_{\max} - \mathbf{s}\_{\min}} \tag{15}$$

$$D(\mathbf{x}) = \frac{d\_{\text{max}} - d(\mathbf{x})}{d\_{\text{max}} - d\_{\text{min}}} \tag{16}$$

where *S*(*x*) is a scaled surrogated output and *D*(*x*) is a scale distance between points evaluated by the objective function. This distance reflects the uncertainty in the estimations of the surrogate model. The minimization of the merit function, min*<sup>x</sup> fm*(*x*), is performed using a random search. The obtained global minimum is then evaluated by the objective function and the result used to update the surrogate model. Now the minimization of the merit function is calculated using the updated model. This process continues for a given number of iterations or until a point is found for which the objective function is below a threshold.

In the case of the drivetrain of the wind turbine, the objective function is defined as the mean absolute percentage error (MAPE) between the active power estimated by the physics-based model and the active power measured by the SCADA system.

$$\text{MAPE} = \frac{100\%}{n} \sum\_{i=1}^{n} \left| \frac{PkW\_i^{sim} - PkW\_i^{real}}{PkW\_i^{real}} \right| \tag{17}$$

Thirteen parameters are involved in the optimization process: four parameters associated with electro-mechanic conversion (electric generator, power converter and wind turbine control), three parameters related to aero-dynamical conversion, three parameters of the control strategy, and finally, three parameters associated with the mechanical drivetrain (Table 3).

The calibration was made in two steps: in the first step, six variables were considered, while in the second step, five more variables were added. Table 4 shows both the initial values defined for each parameter (design value), as well as the values adopted after second calibration (calibrated value).


**Table 3.** Parameters involved in the optimization process.

**Table 4.** Design and calibrated values of parameters involved in the optimization process.


The new values of the calibrated parameters are established, always keeping their physical sense. In fact, an interval with a lower and upper threshold was established for each parameter during the optimization process.

As a result, the mean absolute percentage error (MAPE) between the real active power measured in the SCADA and the value obtained in the simulation using the calibrated models improved from 15% to 2.4% (Figure 10).

**Figure 10.** Generated active power vs. wind speed.

#### *3.2. Failure Hybrid Model of the Use Case*

Once the physic model was calibrated, it was used to simulate the failure conditions. In this use case, the overtemperature in the stator winding was analysed. A thermal circuit was added to the already developed normality hybrid model in Simulink to estimate the temperatures in each phase of the stator winding. It must be considered that the isolation class of the stator winding is a Class F, meaning that it is designed to withstand temperatures of up to 155 ◦C. As shown in Figure 11, this thermal circuit takes into account heat transference generated by the stator currents considering the conduction (between the

winding of each one of the three stator phases) and convection (between the winding of each one of the three stator phases, between each stator winding and the environment and between each stator winding and the rotor). The values of radiation were neglected.

**Figure 11.** Thermal circuit of stator winding.

Conductive heat transfer blocks model heat transfer in the thermal network by conduction through a layer of material. The rate of heat transfer is governed by Fourier's law (18) and is proportional to the temperature difference, material thermal conductivity, area normal to the heat flow direction, and inversely proportional to the layer thickness.

$$\mathbf{Q}\_{\text{cond}} = \frac{k}{s} \, A \, dT \tag{18}$$

Convective heat transfer blocks model heat transfer in a thermal network by convection due to fluid motion (in this case, the air). The rate of heat transfer (19) is proportional to the temperature difference, heat transfer coefficient and surface area in contact with the fluid.

$$\mathbf{Q\_{conv}} = \mathbf{hc} \ A \, dT \tag{19}$$

The inputs that feed the thermal model are the stator currents and the room temperature where the electric generator is installed (in this case the temperature of the nacelle), while the outputs are the temperatures of each phase of the stator winding.

In the real data made available during this study, there are five anomaly cases labelled as overtemperature in the stator winding (Figure 12).

**Figure 12.** Five labelled anomaly cases of overcurrent during real operation (wind speed and active power signals).

The failure modelling was validated using data during these five anomaly cases, obtaining results for the estimated stator winding temperatures, as shown in Figure 13, compared with the real SCADA winding temperature.

**Figure 13.** Temperature values during overtemperature failure-labelled cases.

The MAPE between the real stator winding temperature measured in the SCADA and the value obtained in the simulation using the calibrated model has a value of 11%, with a maximum percentage error of 16% in the worst scenario. This value still has room for improvement if more accurate design data become available for the thermal model.

#### *3.3. Synthetic Failure Data Generation in the Use Case*

A failure model for stator winding overheating was trained with real data from five labelled failures. For this failure mode, four parameters (CIs) were identified: failure or anomaly duration, ambient temperature, nacelle temperature, and wind speed.

The failure duration and ambient temperature are assumed to be uniform during the whole duration of the failure. The distribution of these values in the training data is approximated with a kernel density function (KDE) with a Gaussian kernel (Figure 14). Continuous line represents the probability density functions of the duration and ambient temperature observed in the failure/anomaly instances from the real SCADA, while cross symbols represent real observations This technique, compared with density estimation by histogram, creates a smooth PDF that does not depend on the choice of binning. Instead, a Gaussian component is fitted to each data point. The Gaussian kernel is defined by the function:

$$K(\mathbf{x};h) \propto \exp\left(-\frac{\mathbf{x}^2}{2h^2}\right) \tag{20}$$

where the density function estimated at point *x* of a univariate distribution is:

$$\hat{f}(\mathbf{x};h) = n^{-1} \sum\_{i=1}^{n} \mathcal{K}(\mathbf{x} - \mathbf{x}\_{i}; h) \tag{21}$$

where (*x*1, *x*2, ..., *xn*) are independent and identically distributed random samples from such distribution. The bandwidth *h* is a smoothing parameter that controls the balance between variance and bias in the resulting density function. The resulting Gaussian mixture is a non-parametric estimator of the probability density function able to represent the uncertainty present in a small data sample. In addition, a domain expert can intuitively control the estimator with a bandwidth parameter based on a descriptive analysis of SCADA data and physical properties of the system.

**Figure 14.** Probability density functions (continuous line) of the duration and ambient temperature observed in the failure/anomaly instances from the real SCADA. Cross symbol represents real observations.

The PDF of the wind speed and nacelle temperature variables are dependent on the relative time within a given failure or anomaly. Hence, a generative model aims to learn a PDF from which to sample a time series of a given variable, not simply a single value. Such a function can be approximated by recursively fitting an ordinary least squares (OLS) model to the transition between each time point. In this case, the resulting marginal probability distribution at a given point in time is conditional to the value at the previous time point. The statistical model of the predicted value is:

$$X\_{t1} = X\_{t0}\beta + \varepsilon \tag{22}$$

Additionally, the estimation error *ε* is assumed to have a normal distribution such that:

$$
\varepsilon|X\_{t0} \sim N\left(0, \sigma^2 I\right) \tag{23}
$$

where *σ*<sup>2</sup> is a positive common variance for the elements of the error vector (assuming homoscedasticity) and *I* is the identity matrix.

The generation of random samples starts by the sampling an unconditional seed at time 0. This seed is randomly sampled from a distribution learnt from the training values at time 0. The distribution is approximated by KDE as seen above for the case of ambient temperature. The next data point in the time series, *Xt*1, is sampled from the distribution of *ε* around the prediction mean value *Xt*0*β*. This process iterates for each data point the requested time. Finally, synthetic failure patterns are randomly generated using the learnt statistical distributions (Figure 15) and are fed as inputs into the developed DT.

**Figure 15.** Generation of random patterns (in grey) of wind speed based on real SCADA data (in red).

The DT generates the rest of failure synthetic measurements (e.g., stator winding temperature, and generator output current,) creating a multivariate synthetic failure scenario (Figure 16).

**Figure 16.** Multivariate synthetic failure pattern formed by the output of the data-driven stochastic model and the deterministic functions of the DT.

Figure 17 shows both the synthetically generated stator winding temperature values (in grey), and the stator winding real values measured by the SCADA system (in red). It can be noted that most of the synthetically generated data are similar to the real SCADA data. However, few of the synthetically generated data significantly differ from real data due to the starting seed value.

**Figure 17.** Stator winding temperature calculated by the DT thermal model from synthetic input variables (in grey). Stator winding temperature as measured by the SCADA system (in red).

#### **4. Conclusions and Next Steps**

This paper proposes an approach for creating a hybrid model-based digital twin that combines the benefits of physics-based models with advanced data analytics techniques.

This study has two main innovation outcomes. On the one hand, a process is established to generate synthetic failure data based on real data leveraging different statistical techniques. On the other hand, the process of failure classification based on machine learning techniques, allows anomaly conditions to be identified in the operation of the wind turbine. These two innovations can provide solutions for the main limitations of current digital twin approaches regarding accuracy, explainability, and the lack of sufficient training data.

The synthetic failure data generation process was validated using real operational data from a 1.5 MW power double-fed induction generator wind farm owned by Engie. In more detail, this has been applied to a specific failure (or anomaly) mode, namely the stator winding overtemperature. The obtained results are satisfactory, although further research is necessary. One of the limitations found in current research is the difficulty in achieving detailed labelled failure information.

In future studies, the authors foresee the following research lines. It is envisaged that a developed methodology for failure diagnosis, leveraging non-supervised and supervised machine learning algorithms, could be applied, as explained in Section 2.4. The results of this research could form the basis for future publications, which will likely be derived from the methodology of this article. These algorithms will be trained using real operational data augmented with synthetic failure data generated using this methodology. Furthermore, the authors plan to assess the generalization capacity of the proposed approach, validating it with additional failure modes and other drivetrain technologies (i.e., permanent magnets). Equally, the developed hybrid models might be further improved by applying state-of-theart deep learning techniques. Finally, the scalability of the proposed solution should be assessed by implementing and validating it in an online real-time scenario.

#### **5. Patents**

The work reported in this manuscript is associated with a patent with application number EP22382724.7.

**Author Contributions:** Conceptualization, A.P., E.P. and E.M.; methodology, A.P., E.P. and E.M.; software, A.P. and M.E.; validation, A.P., M.E.; investigation, A.P., E.P. and E.M.; resources, P.C.; writing—original draft preparation, A.P.; writing—review and editing, A.P., M.E., E.P., E.M. and P.C.; visualization, P.C.; supervision, P.C.; project administration, E.M.; funding acquisition, A.P., E.P. and E.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the European Union Horizon 2020 Programme, grant number 872592 (PLATOON-2020), and the Elkartek Programme of Eusko Jaurlaritza KK-2018/00096 (VIRTUAL).

**Data Availability Statement:** Data used in this research are not publicly available due to Engie Green ownership: https://digital.engie.com/en/solutions/darwin (accessed on 4 October 2022). They might be available from the author Phillipe Calvez upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Glossary**



#### **References**


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## *Article* **HVDC Fault Detection and Classification with Artificial Neural Network Based on ACO-DWT Method**

**Raad Salih Jawad \* and Hafedh Abid**

Laboratory of Sciences and Techniques of Automatic Control & Computer Engineering (Lab-STA) Sfax, National School of Engineering of Sfax, University of Sfax, Sfax 3029, Tunisia

**\*** Correspondence: raad.saleh@gmail.com; Tel.: +964-771-338-1461

**Abstract:** Unlike the more prevalent alternating current transmission systems, the high voltage direct current (HVDC) electric power transmission system transmits electric power using direct current. In order to investigate the precise remedy for fault detection of HVDC, this research proposes a method for the HVDC fault diagnostic methodologies with their limits and feature selectionbased probabilistic generative model. The main contribution of this study is using the wavelet transform based on ant colony optimization and ANN to detect the different types of faults in HVDC transmission lines. In the proposed method, ANN uses optimum features obtained from the voltage, current, and their derivative signals. These features cannot be accurate to use in ANN because they cannot give reliable accuracy results. For this reason, first, the wavelet transform applies to the fault and non-fault signals to remove the noise. Then the ACO reduces unimportant features from the feature vector. Finally, the optimum features are used in the training of ANN as faulty and non-faulty signals. The multi-layer perceptron used in the suggested method consists of many layers, enabling the creation of a probability reconstruction over the inputs by the model. A supervised learning method is used to train each layer based on the selected features obtained from the ant colony optimization-discrete wavelet transform metaheuristic method. The artificial neural network technique is used to fine-tune the model to reduce the difference between true and anticipated classes' error. The input signal and sampling frequencies are changed to examine the suggested strategy's effectiveness. The obtained results demonstrate that the suggested fault detection and classification model can accurately diagnose HVDC faults. A comparison of the Support vector machine, Decision Tree, K-nearest neighbor algorithm (K-NN), and Ensemble classifier Machine techniques is made to verify the suggested method's unquestionably higher performance.

**Keywords:** HVDC fault detection; artificial neural network; ACO-DWT; optimization method

#### **1. Introduction**

Due to its lower cost across long distances and capacity to transmit more power, high voltage direct current (HVDC) transmission systems have been widely used for power transmission projects with overhead transmission lines, bulk power, and asynchronous connections. The length of the lines, the surroundings of the transmission lines, and unfavorable weather conditions have all contributed to an increased error rate in HVDC transmission lines [1]. For HVDC transmission lines, current differential protection and direct current (DC) voltage reduction are commonly utilized as backup protection in addition to primary protection systems based on voltage derivatives and traveling waves. Protections based on traveling waves and voltage derivatives are vulnerable to fault resistance because they rely on the pace of the voltage change to identify problems. They frequently misdiagnose high impedance failures [2,3].

When determining a specific fault class, the neural network approaches are growing in popularity among fault prognosis techniques [4,5]. These algorithms require fault features derived from the line data (current and voltage). Even though the fault information was

Academic Editors: Guang Wang, Jiale Xie and Shunli Wang

Received: 27 November 2022 Revised: 10 January 2023 Accepted: 14 January 2023 Published: 18 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

generated from transmission line current or voltage waveforms, a fault class cannot be determined only based on the raw signal data. Then, it was investigated how to use signal processing methods such the wavelet transforms [6], S-transform [7], or Hilbert-Huang transform [8] to separate the important properties that control the behavior of the line faults from the transmission line waveforms. As a result, increasing the neural network-based defect detection and classification model's accuracy has become one of the key study areas.

In [9], a decision tree (DT) based fault detection and classification technique for the microgrid was introduced. The discrete Fourier transform was used to extract the information. To identify the HVDC faults, DT and wavelet transformations were coupled [10]. In grid-tied DG systems, the wavelet transform (WT) and the S-transform can also be used to detect disturbances [11]. To identify the different types of faults, a learning model that combines the naive classifier, support vector machine (SVM), and Extreme Learning Machine (ELM) based on the traits returned by the Hilbert–Huang transform has been utilized [12]. ELM and discrete wavelet transform (DWT) were combined in [13] to detect, classify, and identify a microgrid's sections. For the microgrid's flaw identification and classification, Reference [14] presented a semi-supervised model; first demonstrated in [15], a Taguchi-based artificial neural network (ANN) using DWT. The shallow design constrains these neural networks- and machine learning-based techniques. They employ the ability of the complicated non-linear properties of the HVDC to learn. These methods cannot combine the advantages of numerous aspects with perfection since there are no hidden layers.

Raad Salih et al. [1] used the gray wolf optimization method based on ANN to detect the fault in the HVDC system. They used the gray wolf optimization algorithm to select the best features extracted from the voltage and current signals.

The metaheuristic methods are used in [16,17], and deep learning methods are implemented in [18,19]. The improved power quality and fault detection are presented in [20]. The economic dispatch in the HVDC system is presented in [21]. The protection in sensitive load is investigated in [22,23].

In all approaches mentioned in the literature, feature selection has not been used. Moreover, the wavelet transform cannot give accurate results because some features cannot be reliable for fault detection. It is a significant disadvantage of the previously used methods in the literature. In this study, by combining the WT-ACO, the problem is solved.

This research offers an ANN with numerous layers of hidden units to address the issues and provide a method for learning the intricate non-linear feature of the HVDC in order to increase classification accuracy. ANN was initially used to identify aircraft engine problems. After that, research into diagnosing faults in gearboxes, rolling bearings, and reciprocating compressor valves expanded quickly [24–27]. ANN is a stack of ACO-DWT that deepens the network and makes it possible for the model to extract features in an adaptive manner. ANN can work with non-linear data [28–30]; therefore, it can more precisely classify faults in the microgrid domain.

The proposed network performs fault diagnostics on the HVDC system with phase-tophase and phase-to-ground fault breaker systems using voltage and current waveform data as input. The features are extracted from the raw signal samples using a discrete wavelet transform tool. This research also suggests an extension of ant colony optimization to select the best features created by DWT with the dropout approach to improve the accuracy performance of fault detection. The dropout approach considerably improves the fault detection accuracy performance against a traditional method. The following are the article's primary conclusions:


The protection block begins with the fault detection unit as its first component. Therefore, a quick and dependable solution is required to detect flaws in HVDC protection systems. In this study, an ANN-based on Ant Colony Optimization and wavelet transform is employed to react quickly to detect faults. ANN algorithm requires some time to learn, but after it has completed the learning step, the trained network can move on to the fault detection stage. ANNs may detect faults much faster during the testing phase than traditional logic techniques. The importance of developing fault detection techniques is that they increase accuracy, sensitivity, and reliability, and that is what the transmission authority needs to decrease time and cost for finding and repairing the faults in the HVDC system.

The order of the paper is as follows. The material and method are presented in Section 2. The design of the suggested approach for categorizing HVDC faults and the necessary materials, is also described in Section 2. The performance analysis of the suggested system is shown in Section 3. Section 4 will finalize this paper with a conclusion and future work.

#### **2. Material and Method**

Shunt faults and series faults are the two basic categories under which HVDC power line faults fall [33]. A series fault, also known as a simple break in one or more conductors, occurs when there is an imbalance in the series impedance on the line. Power transmission from one location to another is not directly related to this kind of failure. In contrast, the three-phase power network regularly experiences shunt faults during power transmission, which are subsequently categorized as phase-to-phase (PP), phase-to-ground (PG), and two PG faults (2PG).

A single line-to-ground fault can occur on any phase line of a three-phase power line if it meets the neutral line or hits the ground. The problem brought on by strong winds or trees falling on power lines is also known as a short circuit fault [34]. Figure 1A–C depicts three HVDC system stages and shows three forms of single line-to-ground faults [34].

**Figure 1.** HVDC system fault classes (**A**) a-g, (**B**) b-g, (**C**) c-g, (**D**) ab-g, (**E**) bc-g, (**F**) ac-g, (**G**) a-b, (**H**) b-c, (**I**) a-c, (**J**) a-b-c-g, (**K**) DC fault [34].

A two-PG fault occurs when two lines of a power line fall to the ground. This fault has more asymmetry and a higher fault current amplitude than the line-to-line fault. If this issue is not fixed right away, it could develop into a three-line to-the-ground fault, which is much more dangerous than other fault types. The ab-g, b-g, and a-g faults are depicted in Figure 1D–F, where Rf is the fault resistance [34].

A short circuit between any two lines in a three-phase system causes this kind of failure. One of the significant aspects of this asymmetrical fault is the difficulty in predicting the upper and lower boundaries of the fault impedance due to its magnitude varying over a large range. Three distinct PP fault types are displayed in Figure 1G–I. in contrast, three phases to ground fault appear in Figure 1G. Figure 1K illustrates the DC line to the ground fault. This paper studies the DC faults that may happen in the HVDC system's DC line and AC faults that occur in the AC side of one of two terminal LCC HVDC systems under study.

#### *2.1. System Modelling*

This research proposes a probabilistic, generative network-based framework for detecting and classifying HVDC system faults. The sound or healthy condition was utilized to create a type of fault that provided 11 fault types for the unhealthy or fault detection plan. This sort of fault encompassed all short circuit fault scenarios and the good state of the phase. It was assumed that the classifier's nature would be sound or error-free under typical conditions. A bad or fault event was detected when the classifier output was changed to a particular fault class. Extraction of the fault features from the raw signals was required for the suggested method's training. Each fault signal's energy was unique and determined by the system parameters, such as the fault distance and resistance. The DWT was used to independently assess the variation in each phase's raw signals. After that, each signal's energy was estimated in order to create the necessary dataset. Figure 2 shows the HVDC system under test, and Table 1 includes the parameters of the system Voltage Source Converters (VSC) that are used in modern HVDC; nevertheless, the model used in this study makes use of thyristors. The literature contains well-known and cutting-edge protection techniques for thyristor-based two-terminal HVDC systems [35–37]. VSC-based systems, particularly multi-terminal DC systems, are currently facing protection issues. The GWO approach, the study's main topic, is used to assess the features and choose the appropriate voltage and current signal format.

**Figure 2.** LCC-HVDC system under study.

**Table 1.** Parameters of the HVDC system under study.



#### **Table 1.** *Cont.*

#### *2.2. Effect of Fault Distance on Signal Energy*

A defective occurrence could happen anywhere along the HVDC systems. The proposed ANN's training method, which varied the network, could inspect the signal across the entire HVDC system thanks to the fault distance. In order to create the sample data, the fault event's position was changed from the current and voltage waveforms, and the distance measurements ranged from 1 to 19 km with a 0.5 increment. The change in the original signal produced different signal energy lengths, which ultimately represented different features.

#### *2.3. Wavelet Transform for the Generation of Fault Feature*

In this work, the qualities of a particular section of a signal during a quick change in the signal were examined using the DWT. The WT works by breaking down a signal into a sequence of temporal components. A time series waveform receives significant guidance from the time series faulty sections, which shield a specific frequency range. Here is an example of how the WT approach represents, during a fault event, the fault attribute of a particular piece of the faulty signal. a group of low-pass (LPF) and high-pass (HPF) filters are used in DWT to process signals (LPF). The signal is broken up into detail (Det) and approximation (App) coefficients as a result of the LPF's analysis of the examination of the high-frequency domain signal by the HPF and the low-frequency domain signal. The fault signal's large- and small-scale frequency components are represented utilizing the App coefficient. Therefore, the fault signal is represented by its small- and large-scale frequency components by the Det coefficient. The subsequent App replicates this decomposition procedure, dividing a fault signal into several lesser-resolution pieces.

#### *2.4. Model for Proposed Hierarchical Generative Faults*

This section introduces the ANN framework for HVDC fault detection and classification. An arrangement of constrained machine learning makes up the proposed ACO-DWT. Using the ACO-DWT, the features are extracted and selected accurately.

#### *2.5. Wavelet Analysis*

A wavelet transform function is the display of lower-frequency signals larger and highfrequency signals narrower when wavelet detection is present. Despite similarities, there are a few significant differences between the Fourier transform and the wavelet transform. The signal is divided into sines and cosines via the Fourier transform. On the other hand, the wavelet transform can be used with elements in both Fourier and real spaces. The temporal widths of the wavelet transform can be changed to match the frequency. This attribute of frequency width auto-tune is most helpful when assessing electromagnetic transients that have superimposed on the frequency power components are high-frequency components [38]. Typically, the wavelet transform looks like this:

$$WT(f, a, b) = \frac{1}{\sqrt{2}} \int\_{-\infty}^{\infty} f(x) \psi^\* \left(\frac{t - b}{a}\right) \tag{1}$$

where *a* and *b* are the function constants, which are also known as the scaling and translation parameters, and (\*) is the complex conjugate of the wavelet function *ψ*. Continuous wavelet transforms (CWT), and discrete wavelet transform are the two subcategories of the wavelet transform (DWT). The wavelet transform is derived into the correlated wavelet transform (CWT), which uses redundant wavelets and arbitrary scales. By breaking down the signal into orthogonal sets using a discrete set of wavelet scales, the discrete wavelet transform (DWT) is produced. The discrete wavelet transform is obtained using the following expression (DWT).

$$DWT(f, m, m) = \frac{1}{\sqrt{a}} \sum\_{k} f(k) \psi^\* \left(\frac{n - ka\_0^m}{a\_0^m}\right) \tag{2}$$

The parameters "*a*" and "*b*" are swapped out for *a<sup>m</sup>* <sup>0</sup> and *ka<sup>m</sup>* <sup>0</sup> , where *k* and *m* are integers compared to the term 2.17. The DWT functions as a bank of low-pass and highpass filters that provide low-pass and high-pass subbands for the signal. The low-pass subband is subjected to the same procedure to create narrower low-pass and high-pass sub-bands. Wavelet transforms, either continuous or discrete, can be used to assess the estimated distance to the fault. In order to define a mother wavelet from a voltage transient waveform, a continuous wavelet transform is utilized in the research of fault location in power networks [39]. However, the analysis of this study can give good results with just the discrete wavelet transform (DWT). Figure 3 shows the wavelet family featuring the daubechies3 (db3) wavelet mother, which is utilized to decompose voltage waveforms registered by DFRs into its five coefficients (WTCs). Due to their abundance of highfrequency content, the WTCs at level 1 (D1) are subsequently analyzed to identify the times of arrival (ToAs). These signals are finally squared to create WTC2, as was conducted in [38], to reduce noise in WTCs.

**Figure 3.** Wavelet Daubechies' family.

The signals with and without faults were constructed to investigate the HVDC fault detection in this work. Multiple signals with various AC and DC fault types were devised for this purpose. These signals are used to extract the 12 properties that depend on the voltage, current, and their individual components. Some of these features are inappropriate for ANN training [18,19], and employing them will lead to mistakes and lower detection accuracy. The best and most precise characteristics ought to be chosen for this purpose. Thus, the feature selection uses the ACO approach, which was first introduced in [40]. Figure 4 depicts a summary of the suggested technique.

**Figure 4.** Summary of the suggested technique for finding HVDC faults.

As shown in Figure 4, Simulink-MATLAB manually generates the 11 faulted signals, including the AC and DC faulted signals. Next, the output signals are obtained for each fault. The neural network may occasionally make errors in fault identification because these signals contain many characteristics, the majority of which are unsuitable for training. Therefore, the best and most practical characteristics must be chosen to train the network. The best features are selected using the ACO approach, and the neural network is trained using these features. On the right side of the flowchart is the ACO method scenario.

The main objective of this study is to use the optimum features of the fault and nonfault signals by using the Wavelet transform based on the ACO algorithm. The wavelet transform removes the noisy signal from the current and voltage signals obtained from the system. Furthermore, ACO uses to find the optimum features that affect the training of ANN to recognize the fault and non-fault signals.

First, a visual representation of every feature in the S dataset is presented. All nodes are connected to one another and are referred to as nodes. The number of ants and the number of repetitions should then be determined [41]. The value is known as the pheromone trail, and all of its values are initially set to a fixed value of one at the beginning of the algorithm. also known as the value of heuristic information, is equal to the reciprocal distance between the qualities [42], which will be determined in this article using the two approaches, FC and FF.

The algorithm is usable after establishing the initial values. The ant is initially placed on a node at random in each iteration. The rule of transfer is applied to derive the following ninety, as indicated in Equations (3) and (4):

$$P\_i^k(t) = \frac{|\tau\_i(t)|^a \* \times |\eta\_i(t)|^\beta}{\sum\_{a \in \mathcal{A}} |\tau\_i(t)|^a \times |\eta\_i(t)|^\beta} \quad if(q > q\_0) \tag{3}$$

$$\dot{q} = \max\_{u \in \mathbb{R}^k} (\tau\_i(i)^a \times \eta\_i(i)^\beta) \qquad \text{if } (q < q\_0) \tag{4}$$

The *α* and *β* values are chosen in order to increase the efficacy of *τ* and *η*. The ant has not yet encountered the attributes in *j <sup>k</sup>*, and the only trait they have is zero. The parameter *q*0, whose value is a random number between 0 and 1, is crucial in selecting whether to use the greedy or probabilistic approach.

The amount of pheromone collected from the scan should be updated in accordance with Equation (5) when the n ant has finished the node scan:

$$
\pi\_i(t+1) = (1 - \rho)\pi\_i(t) + \sum\_{i=1}^n \Delta \pi\_i^k(t) \tag{5}
$$

To lessen the effect, the value of the average number of nodes chosen for the Filter technique is equal to Δ*τ<sup>k</sup> <sup>i</sup>* is determined, which is the reverse of the error achieved using the Wrapper technique [41,42].

#### *2.6. Criteria for Distance or Similarity of Features*

Two types of relationships exist between two random variables: linear and non-linear. The correlation coefficient formula is the most well-known formula for calculating linear variables. To compute non-linear variables, they employ information theory and the entropy approach. The correlation coefficient technique has the drawback of being ineffective with batch and non-numerical data, but the entropy method performs well [43].

A discrete or continuous random variable's uncertainty is measured using entropy or irregularity criteria. The discrete random variable *X* = (*x*1, *x*2,..., *xn*) has an entropy of *H*(*X*) that is determined using Equation (6).

$$H(X) = -\sum\_{i=1}^{n} p(\mathbf{x}\_i) \log(p(\mathbf{x}\_i)) \tag{6}$$

where *p*(*xi*) is the probability value of *xi* happening on the entire set.

In accordance with Equation (7), the two discrete random variables' entropies should be calculated *X* and *Y*.

$$H(X,Y) = -\sum\_{i=1}^{n} \sum\_{j=1}^{n} p(\mathbf{x}\_i, y\_i) \log(p(\mathbf{x}\_i, y\_i))\tag{7}$$

The conditional entropy of *X* to condition *Y* is determined using Equation (8).

$$H(X|Y) = -\sum\_{i=1}^{n} \sum\_{j=1}^{n} p(\mathbf{x}\_i, y\_i) \log(p(\mathbf{x}\_i \middle| y\_i)) \tag{8}$$

The aforementioned formulas' goal is to determine the information factor (IF). The IF criterion, which is in agreement with Equation (9), is utilized to analyze how dependent the two variables are:

$$I(X,Y) = H(X) - H(X|Y) \tag{9}$$

The two variables are independent if the value of IF is zero, and the larger this value, the more dependent *X* and *Y* are [44]. The correlation between the information coefficient and entropy is depicted in Figure 5.

**Figure 5.** Relationship between information coefficient and entropy.

The symmetrical uncertainty (SU), or normalized form of IG, used in this work is compatible with Equation (10). This formula's benefit is the normalcy of the two variables' dependence between 0 and 1. The two variables are dependent if the value of SU is close to one and independent if the value of SU is close to zero.

$$SLI(X,Y) = \frac{2 \ast I(X,Y)}{H(X) + H(Y)}\tag{10}$$

Two criteria *SUFC* and *SUFF* are employed in this paper to calculate the *η* [43].

The reliance of each attribute on the class is the definition of the *SUFC* criterion. The more vital and desirable that characteristic will be, the closer this number is to one.

$$\eta\_i = \frac{1}{1 - S \mathcal{U}\_{\text{FC}}} \tag{11}$$

The term *SUFF* refers to the interdependence of two qualities. If its value is very near to one, it indicates that the two traits are quite comparable, and we might consider eliminating one of the features.

$$\eta\_i = \frac{1}{SLI\_{FF}}\tag{12}$$

When choosing attributes, we try to keep class-related attributes and remove redundant or unnecessary attributes. The objective is to select features that have *SUFC* higher and *SUFF* lower values [45].

In the first step of this research study, the voltage and current signals are generated. Then these signals are analyzed to determine the characteristics of the voltage and current signals. The ACO-DWT approach is utilized to choose the best and most effective characteristics. In order to identify the most useful features with which to train ANN, the proposed method based on the ACO algorithm is used.

#### **3. Results and Discussion**

This section uses various parameter adjustments to show how well the suggested fault detection and classification technique performs. Different magnitudes were disclosed by the fault current and voltage signals in grid-connected and islanded modes. As a result, creating a uniform fault classification scheme was challenging. As a result, the effectiveness of the suggested strategy was examined individually under different operating modes and system topologies. Three factors were taken into consideration when determining the accuracy:


In machine learning, a list of data samples is used to test the model's performance and should be different from the training data in order to determine how effective a learning model is. The current and voltage waveforms for each dataset were divided into 1716 samples, which were then combined and shuffled before being randomly selected to assess the effectiveness of the suggested strategy. When the 11 distinct fault classifiers were entered into the 11 × 11 matrix's *x*- and *y*-axis, the confusion matrix (CM) was used to simulate the performance of ANN for Lines 1–3 under various system configurations and HVDC operating modes. The vertical levels represent the projected fault class, while the horizontal levels indicate the actual class. The true positive (TP), true negative (TN), false positive (FP), and false negative (FN) counts are also reported in the confusion matrix and are defined as:

TP: The classifier correctly predicted a label and is a member of the original class;

TN: The classifier successfully predicts a label even when it does not fall under the initial category;

FP: The classifier predicts a label to be positive even when it does not belong to the original category;

FN: The label that the classifier predicts would be negative but belongs to the original class.

First and foremost, according to the CM, most of each system configuration's fault classes were assigned correctly. The classification accuracy of the average confusion matrix (Acc) was used as the first accuracy measurement criterion Equation (13).

$$Accuracy = \frac{(TP + TN)}{(TP + FP + TN + FN)} = \frac{N\_{TD}}{N\_{CC}} \tag{13}$$

Here, *NTD* displays the total number of data used to create the model. *NCC* denotes the number of correctly categorized data. The remainder of the HVDC system might function similarly to the proposed work.

According to the findings, the proposed classifier's grid-connected radial mode operation had the best accuracy, which was 99.70%. The classifier performed better than 99.5% for the other system settings, which was as expected. The Confusion matrix for different classifiers, SVM, Decision Tree, K-NN, and Ensemble method, is shown in Figure 6.

The average accuracy for various system setups is shown in Figure 7.

**Figure 6.** Confusion matrix for different classifiers (**A**) SVM, (**B**) Decision Tree, (**C**) K-NN, and (**D**) Ensemble.

The average accuracy, however, was unable to provide a complete analysis of the model's performance. The sensitivity, specificity, and accuracy were then used to assess the classification performance in order to determine how the classifier handled different fault types. The mentioned criteria are regarded as ideal when it equals one and worst when it equals zero. The sensitivity, also referred to as the positive predictive value, is defined as follows:

$$Sensitivity = \frac{TP}{(TP + FN)}\tag{14}$$

A good classifier should have a precision value of one. From Equation (14), the precision value declines as the FP rises, which is unexpected for a strong classifier. Specificity, another statistic that is also referred to as the true positive rate or the classifier's sensitivity, is defined as follows

$$Specificity = \frac{TN}{(TN + FP)}\tag{15}$$

The best classifier's sensitivity value should be 1, just like the precision. For this metric, the recall value fell as the FN grew, which was also contrary to expectation. As a result, accuracy, which considers both true positive and true negative, was used as another performance evaluation indicator. Given that Table 2 shows the voltage and current signals, the recommended classifier's greater accuracy demonstrated that it had fewer false positives and negatives. Additionally, each fault class's categorization accuracy (user accuracy) made it clear that the classifier had a high degree of accuracy in its capacity to categorize the problems.

**Figure 7.** Average accuracy for various system setups.


**Table 2.** Accuracy, sensitivity, and specificity of the proposed classifier.

Figure 8 is a graphical depiction of these findings.

**Figure 8.** Graphical chart illustration.

Due to the lack of both signals at the same time, the system must perform the categorization tasks using voltage or current waveforms. Different input signal types and sampling rates were employed to examine the fault classification performance of the proposed classifier. The input signal types used in this inquiry were the voltage waveform, the current waveform, and the combined current and voltage waveform. The SF used were 2, 5, 10, 15, and 20 kHz. Five times through the classification process, the findings for an SF and a certain signal type were determined. To obtain the final findings presented in Figure 9, the mean value of the accuracies was calculated.

The improvement in classification accuracy was anticipated since a short circuit fault class with a greater SF carries more specific fault information. The three-phase current waveform performed better for classification at lower sample rates than the three-phase voltage waveform. This scenario was anticipated in part because, for a given fault class, compared to the current waveform, the voltage waveform contained less information about low-frequency faults.

On the other hand, the voltage waveform had a few spare incorrect transients that could be used to examine the short circuit fault's specifics more thoroughly. The aforementioned analysis suggests that relying just on current or voltage waveforms would not produce the requisite precision. Given that both the three-phase current and voltage intentions utilized specific short circuit fault information, if both waveforms were taken into account simultaneously, the designated frequency level may be obtained with a higher

fault categorization performance. The study's results showed that using only the current or voltage waveforms to classify data yielded poor results; however, their fusion produced a classification accuracy of more than 99% at the high-frequency range level taken into account, validating the efficacy of the proposed FDC model. The remaining HVDC system under evaluation can show similar categorization results. Table 3 compares the accuracy, sensitivity, and specificity of the entire fault classification system for the HVDC fault classification with the suggested technique and alternative methods.

**Figure 9.** Accuracy of the classification for the proposed classifier.


**Table 3.** The proposed method's accuracy, sensitivity, and specificity.

The best aspects of voltage, current, and derivatives are utilized in the suggested strategy. It is contrasted with the multi-layer perceptron (MLP), radial basis function (RBF), learning vector quantization (LVQ), and self-organizing map (SOM) neural networks. Results from the studies showed that the advised approach, ANN, RBF, LVQ, and SOM had accuracy values of 98.86, 98.65, 98.78, 98.30, 99.49, 99.60, 99.48, and 99.45, respectively. The proposed method had the highest accuracy because when the feature selection component

accuracy rose to 99.60%. The graphical illustration of the results is shown in Figures 10 and 11.

of the ACO-DWT algorithm, which is based on the Decision tree classifier, was utilized, the

**Figure 10.** Graphical illustration of the accuracy, sensitivity, and specificity.

**Figure 11.** Graphical illustration of the TP, TN, FP, and FN.

#### **4. Conclusions**

In order to identify the optimum characteristics of the voltage and current signal to utilize in ANN to train the system, this paper suggested the ACO for feature selection. For the grid-connected and island modes of the HVDC, the ACO approach is robust in identifying the best aspects of the signals to detect and classify the faults. The suggested approach ensured that the model automatically recognized and analyzed abnormal signals pertaining to various HVDC failures. This was accomplished by measuring the voltage and current waveforms separately and utilizing feature extraction to compare them to variations in the line characteristics. The proposed method's usefulness as a generalized model that worked at different sampling frequencies was confirmed by using both the current and voltage parts for fault diagnosis and classification. The suggested technique's efficiency was assessed using various experiments, such as those examining the influence of signal type. The results demonstrated that the suggested fault detection and classification model correctly recognized and categorized short circuit faults for all fault categories with an accuracy close to 99.60%. The best scenario was obtained from the ANN-DWT-ACO-DT method, with 99.60, 99.25, 99.96, 99.62, 98.14, 0.04, and 0.75 for accuracy, sensitivity, specificity, TP, TN, FP, and FN, respectively. Using both the voltage and current waveforms within the tested frequency range showed the model's impressive performance. The authors advise employing the Fourier transform for feature extraction of the current and voltage signals and various metaheuristic techniques to identify the accuracy rates for defect detection situations.

**Author Contributions:** Conceptualization, R.S.J. Methodology, R.S.J. Software R.S.J. Validation, H.A. Investigation R.S.J. Writing—original draft preparation, R.S.J. Writing—review and editing, H.A. Visualization, R.S.J. Supervision, H.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors declare that there was no funding for this work.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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**Yifeng Lin 1,\*, Jingfu Gan <sup>2</sup> and Zengping Wang <sup>1</sup>**


**Abstract:** In modern power systems, the installation of a shunt capacitor bank is one of the cheapest and most widely used methods for improving the voltage profile. One shunt capacitor bank is composed of mass capacitor units and have ground, ungrounded, delta, wye connections that make configuration of capacitor banks is various. In the case of long-term operation, the failure of a single capacitor unit of a capacitor bank is likely to cause uneven voltage, which will lead to the breakdown and burning of the whole group, resulting in huge losses. The relay protection device can detect the simultaneous voltage and current of the capacitor. By utilizing these data from the relay, the abnormal state of the shunt capacitor banks at the initial stage of the fault can be found through monitoring the slight change in capacitance. Timely and early maintenance and repair would avoid capacitor bank faults and potentially greater economic losses. Capacitor banks have different connection modes. For ungrounded wye-connected capacitor banks with an unknown neutral point voltage, the capacitance parameters of each branch cannot be calculated. A parameter symmetry based on the calculation method for capacitor parameters is proposed. For long-term monitoring and observation of the capacitor capacitance value, the fault state and abnormal state of the capacitor are identified based on statistical methods. The simulation established by PSCAD verified that a relay protection device can realized an effective monitoring of the early abnormal state of the capacitor bank.

**Keywords:** shunt capacitor fault; equivalent balance equation; capacitance value calculation; capacitor monitoring

#### **1. Introduction**

In the modern power system, capacitors are widely used in energy storage, voltage regulation, filtering and other scenarios due to their simple structure and limited manufacturing and maintenance costs [1]. As a reactive power supply, shunt capacitors can adjust the system voltage, improve the power quality and reduce the line loss [2]. As their implementation has increased, shunt capacitor banks have become one of the power devices with the highest failure rate. Faults in capacitor banks have caused group explosions and group damage many times [3,4], resulting in significant fluctuations in grid voltage, increasing active and reactive power losses, reducing the service life of capacitors and compromising the safety of the power grid.

The literature includes a great deal of in-depth research on the use of protection and monitoring technology of shunt capacitors to improve reliability and reduce losses. The most common means of protecting capacitors is to use different connection and voltage levels, with an emphasis on configuration protection [5–12] References [5,6] at the 500 kV voltage level, parallel compensation of substations and lines and series compensation capacitors. In addition, the protection of these capacitors is analyzed in detail, and the optimal protection configurations and scheme setting principles are given for each type of capacitor. Reference [7] calculates and analyzes the sensitivity and settings of relay protection under the various modes of the shunt capacitor banks in the 1000 kV ultra-high

**Citation:** Lin, Y.; Gan, J.; Wang, Z. On-Line Monitoring of Shunt Capacitor Bank Based on Relay Protection Device. *Energies* **2023**, *16*, 1615. https://doi.org/10.3390/ en16041615

Academic Editors: Guang Wang and Abu-Siada Ahmed

Received: 15 November 2022 Revised: 31 January 2023 Accepted: 3 February 2023 Published: 6 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

voltage (UHV) power system in China. Reference [8] introduces the setting principle for unbalanced protection of the H-bridge high-voltage capacitor banks. These methods provide excellent protection, but the specialized protection design and settings need to be based on different voltage levels and wiring forms, which are more complicated and costly; these need to be studied separately according to the actual working conditions of the system. Another research method relies on the development of smart substations to realize the monitoring function of shunt capacitors [13–16]. The capacitor fault monitoring system in [13] extracts synchronous voltage and current signals through specialized devices to monitor shunt capacitor banks in real time, which requires more space and is more costly. Reference [14] uses compensated negative sequence and neutral currents to locate internal faults and can locate faults that occur simultaneously in either of the two branches of a double-wye parallel capacitor bank. Reference [15] proposes a new scheme based on the unbalanced current at the neutral point. By calculating the unbalanced current distribution under all possible operating conditions, the fault severity and fault location can be identified. Reference [16] designs a very sensitive real-time capacitor-monitoring device based on calculating capacitance through the variation of LC oscillation frequency for early internal component failures of capacitor banks. This type of method mostly aims to design a dedicated monitoring device in an intelligent substation system that effectively reduce the high failure rate of capacitors and be highly universal.

Because the second method mentioned above is more economical and practical, this paper proposes a new monitoring method for shunt capacitors. The relay protection device installed on the bus can easily obtain the simultaneous voltage and branch current of the shunt capacitor bank (the voltage is equal to the bus voltage, and the branch current of the capacitor can be obtained from different CT taps). It will be very economical and convenient to only use this information to monitor the shunt capacitor banks. However, it has difficulty calculating the ungrounded wye configuration when there is no additional device to obtain the neutral voltage. In this case, the equations are solved by using the additional conditions of symmetric parameters of two branches and setting alarms according to guidelines and statistical methods that prevent capacitor faults and potential losses. An on-line monitoring method for shunt capacitor banks that is not affected by the connection method is constructed. The simulation verifies the effectiveness and feasibility of this method.

#### **2. Principles of Shunt Capacitor Bank Monitoring**

There are many types of shunt capacitor bank faults. When the internal capacitor fault is caused by overvoltage, harmonics, product defects, etc., the capacitance change value mostly occurs during the initial stage [17]. Factors such as overvoltage and harmonics cause abnormalities and failure of the individual capacitor unit in the internal series-parallel components. Damage to a single component compromises the operating state of other components; this situation gradually evolves and ultimately leads to the failure of the high-voltage shunt capacitors.

In the initial stage of capacitor fault, the abnormality of an individual unit of the shunt capacitor bank can be identified via variation in capacitance. The on-line monitoring method of the shunt capacitor bank can be realized based on the simultaneous voltage and current obtained by the relay protection device at the bus of the shunt capacitor [18].

The most common internal capacitor fault is the breakdown of internal capacitor units. There are three kinds of breakdown faults: electric breakdown, thermal breakdown and partial discharge breakdown. Electric breakdown is mainly due to the rapid breakdown of defective capacitors due to high voltage, high harmonics and other factors. Electric breakdown occurs over a short period of time, the relationship with environmental factors is small, maintenance cannot be performed in a timely manner and monitoring is of little significance. The other two kinds of breakdown develop gradually, which will produce a dielectric change before the breakdown fault. The fault can be found by measuring the capacitance.

According to the guide for the protection of shunt capacitor banks, there are three kinds of fuse protection: internal fuses, external fuses and fuseless. Internal fuses offer very effective protection. When a unit fails, the overcurrent causes the fuse to blow. If a single faulty unit is isolated, the shunt capacitor can continue to operate. However, if an internal fuse exhibits the failures shown in Figure 1b, the group capacitors with the faulty unit will be short-circuited, and the remaining capacitor groups will operate in overvoltage mode. Over a long period of time, a more serious fault would occur. Timely detection of changes can effectively avoid potential losses. If the faulty unit of an internal fuse is isolated as shown in Figure 1c, the shunt capacitor bank with a single faulty unit can still operate normally; the capacitance value changes, however, resulting in uneven voltage across the group. Long-term operation may also cause other unit faults, even serious failures. Timely detection and replacement of faulty parts can also effectively prevent the expansion of a fault.

**Figure 1.** Shunt capacitor bank with internal fuses.

To sum up, most internal capacitor faults undergo a long process of capacitance value change. This process is affected by environment, voltage level and other factors, and it is impossible to formulate unified rules for monitoring. However, the capacitor has been comprehensively inspected at the initial stage of installation and the failure rate is low; this will be taken as the normal state at this time. By storing the data at this time and comparing real-time data with statistical methods, it can be determined whether there is a significant change. If there is a significant change, the capacitor is considered abnormal. The statistical method is general, and the reference sample is its own normal state sample; it can therefore be applied to most of the shunt capacitor banks that operate for a long time (data support).

The connections in shunt capacitor banks are wye, delta and double-wye connections. The relay protection has the following types: Zero-sequence voltage protection performs well for shunt capacitor grounding faults. Differential protection applies to all capacitor external faults. Overcurrent protection is the basic protection for all types of capacitors [19]. Double-wye connection has extra overcurrent protection at the neutral point. This connection is also divided into two cases: ungrounded and grounded [20]. The voltage at both ends of the grounding capacitor is equal to the bus voltage. The capacitance value of each phase can then be calculated by obtaining the branch current [21]. However, an ungrounded capacitor cannot calculate the capacitance value in this way, as it requires additional equipment to extract the neutral point voltage. In this paper, no additional equipment is required for calculating the ungrounded shunt capacitor bank. The simultaneous voltage and current data from the bus are the only data used to calculate the capacitance value as the monitoring criterion for the capacitor bank.

#### *2.1. The Method of Calculating Capacitance Value*

The four most common configurations of a wye-connected capacitor bank are shown in Figure 2. The CT/PT of the relay protection is built on the bus. The grounded-connected line parameters can be can easily calculated based on three-phase voltage and current, after which the capacitance change can be observed via the change in the line parameters. The calculation of ungrounded-connected line parameters is needed to obtain the voltage of the neutral point. However, increase of PT circuits means more cost and lower reliability.

**Figure 2.** Most common capacitor bank configurations.

Ungrounded capacitor banks mainly consist of the wye and double-wye connections. Both wye and double-wye connections can be simplified as shown in Figure 1 [4]. We look at each phase capacitor as an impedance and we monitor its change. The change in the phase capacitance reflects the operating state of the capacitor. The advantage of monitoring the capacitance of the capacitor is that, compared with the unbalanced protection of the capacitor, the monitoring amount is the capacitance value, which is more intuitive and can better reflect the status of the capacitor. Compared with regular maintenance, monitoring is simpler and requires less time [22]. The capacitor circuit is equivalent to that shown in Figure 3, and the solution process is as follows:

**Figure 3.** Capacitor equivalent circuit.

An ungrounded double-wye connection can be simplified as shown in Figure 2. The ungrounded wye and delta connections can be simplified as shown in Figure 2c. . *Uk* (*<sup>k</sup>* <sup>=</sup> *<sup>A</sup>*, *<sup>B</sup>*, *<sup>C</sup>*) and . *Ik* (*k* = *A*, *B*, *C*) are the bus voltages and branch currents of the capacitors. . *ZA*, . *ZB* and . *ZC* are the equivalent three-phase-impedance values, including parallel

capacitors, line impedances and series reactors. The two-branch-current equation is shown in the following [23].

$$\begin{cases} \dot{\mathcal{U}}\_A - \dot{\mathcal{I}}\_A \dot{\mathcal{Z}}\_A = \dot{\mathcal{U}}\_B - \dot{\mathcal{I}}\_B \dot{\mathcal{Z}}\_B\\ \dot{\mathcal{U}}\_B - \dot{\mathcal{I}}\_B \dot{\mathcal{Z}}\_B = \dot{\mathcal{U}}\_C - \dot{\mathcal{I}}\_C \dot{\mathcal{Z}}\_B \end{cases} \tag{1}$$

. *ZA*, . *ZB* and . *ZC* are unknown quantities. Because there are only two equations in Equation (1), it cannot be solved. An additional equation is therefore required to solve the equations.

In the normal state, the three-phase parameters are equal. In the case of a single-phase fault or abnormal state, the equivalent impedance of the two other normal phases are the same. In the case of a two-unit fault in a different phase, the same working condition, model and operating conditions are assumed, so that the changes of the two-phase faults are similar. In the case of a three-phase capacitor fault, the parameters remain approximately equal for the same reason. To sum up, the operating characteristic of capacitors is that at least two-phase have the same impedances. The shunt capacitor bank must have the same three-phase capacitance when installed. Depending on the operating characteristics of the capacitor, two-phase parameters are set equal in turn, i.e., . *ZA* <sup>=</sup> . *ZB*, . *ZA* <sup>=</sup> . *ZC* and . *ZB* <sup>=</sup> . *ZC*; these are then combined with Equation (1) to obtain Equation (2) as follows:

.

$$\begin{cases}
\dot{\mathcal{U}}\_A - \dot{\mathcal{I}}\_A \dot{\mathcal{Z}}\_A = \dot{\mathcal{U}}\_B - \dot{\mathcal{I}}\_B \dot{\mathcal{Z}}\_B \\
\dot{\mathcal{U}}\_B - \dot{\mathcal{I}}\_B \dot{\mathcal{Z}}\_B = \dot{\mathcal{U}}\_C - \dot{\mathcal{I}}\_C \dot{\mathcal{Z}}\_B \\
\dot{\mathcal{Z}}\_A = \dot{\mathcal{Z}}\_B \text{ or } \dot{\mathcal{Z}}\_B = \dot{\mathcal{Z}}\_C \text{ or } \dot{\mathcal{Z}}\_B = \dot{\mathcal{Z}}\_C
\end{cases} \tag{2}$$

$$\begin{aligned} \text{When } \dot{Z}\_A = \dot{Z}\_{B'} \text{ we obtain the solution } \begin{cases} \dot{Z}\_{A1} \\ \dot{Z}\_{B1} \\ \dot{Z}\_{C1} \end{cases} \\ \text{When } \dot{Z}\_B = \dot{Z}\_{C'} \text{ we obtain the solution } \begin{cases} \dot{Z}\_{A2} \\ \dot{Z}\_{B2} \\ \dot{Z}\_{C2} \end{cases} \\ \text{When } \dot{Z}\_B = \dot{Z}\_{C'} \text{ we obtain the solution } \begin{cases} \dot{Z}\_{A3} \\ \dot{Z}\_{B3} \\ \dot{Z}\_{C3} \end{cases} \end{aligned}$$

In order to simplify the calculation, three-phase voltages are set symmetrically and the three-phase voltages are . *UA*, . *UB* and . *UC*. Under normal working conditions, threephase line parameters are also symmetrical. . *UA*, . *UB*, . *UC*, . *IA*, . *IB* and . *IC* therefore have the following relationship:

$$
\dot{\mathbf{U}}\_B = \dot{\mathbf{U}}\_A \angle -120^\circ \,\,\,\dot{\mathbf{U}}\_C = \dot{\mathbf{U}}\_A \angle 120^\circ \tag{3}
$$

$$
\dot{I}\_B = \dot{I}\_A \angle -120^\circ \text{ } \dot{I}\_C = \dot{I}\_A \angle 120^\circ \text{ } \tag{4}
$$

when the assumption condition is consistent with the actual situation, the solution resulting from the state equation is correct, and when the assumption condition is not consistent with the actual situation, the solution resulting from the state equation is incorrect.

Assuming that phase A fails, then analysis proceeds via the variable method. When the capacitance value of phase A decreases, the three-phase currents become . *IA* <sup>+</sup> . *λa*, . *IB* and . *IC*, where . *IA*, . *IB* and . *IC* are still three-phase symmetrical, and . *<sup>λ</sup><sup>a</sup>* and . *IA* are in the same direction.

<sup>1</sup> When it is assumed that phase B and phase C impedances are same (that is, the actual condition), the results are

$$\begin{cases} \dot{Z}\_{A1} = \frac{1}{\dot{I}\_A + \dot{\lambda}\_d} (\dot{I}\_A - \dot{U}\_B + \dot{I}\_B \frac{\dot{I}\_B - \dot{U}\_C}{\dot{I}\_B - \dot{I}\_C})\\ \dot{Z}\_{B1} = \dot{Z}\_{C1} = \frac{\dot{U}\_B - \dot{U}\_C}{\dot{I}\_B - \dot{I}\_C} \end{cases} \tag{5}$$

It is known that, in this case, the calculation result is correct, the capacitance value change of phase B and phase C is 0, the impedance of phase A declines most—that is, the change amount is the largest—and . *ZA*<sup>1</sup> is used for comparison.

<sup>2</sup> When it is assumed that the parameters of phases A and B are equal (not the actual condition), the results are

$$\begin{cases} \dot{Z}\_{A2} = \dot{Z}\_{B2} = \frac{\dot{\mathcal{U}}\_{A} - \dot{\mathcal{U}}\_{B}}{\dot{I}\_{A} + \dot{\lambda}\_{A} - \dot{I}\_{B}}\\ \dot{Z}\_{C2} = \frac{1}{\dot{I}\_{C}} (\dot{\mathcal{U}}\_{C} - \dot{\mathcal{U}}\_{B} + \dot{I}\_{B} \frac{\dot{\mathcal{U}}\_{A} - \dot{\mathcal{U}}\_{B}}{\dot{I}\_{A} + \dot{\lambda}\_{A} - \dot{I}\_{B}}) \end{cases} \tag{6}$$

Equations (3) and (4) can be substituted into (5) and (6), and the three-phase impedance values are compared in <sup>1</sup> and <sup>2</sup> , respectively. In this case, . *ZA*<sup>1</sup> is similar to . *ZC*2, and all vectors in . *ZC*<sup>2</sup> are rotated 120 degrees clockwise; the molecular part is enlarged several .

times, yielding the following: 1 . *IA*<sup>+</sup> . *λa* ( . *UA*<sup>−</sup> . *UB*+. *IB UB*<sup>−</sup> . *UC* . *IB*−. *IC* ) 1 . *IA* ( . *UA*<sup>−</sup> . *UC*+. *IC* . *UB*<sup>−</sup> . *UC* . *IB*<sup>+</sup> . *<sup>λ</sup>a*−. *IC* ) < . *IA*+ √3 3 . *λa* . *IA*<sup>+</sup> . *λa* <sup>&</sup>lt; 1. That means .

*ZA*<sup>1</sup> <sup>&</sup>lt; . *ZC*2. The group with the largest impedance variation is thus the actual condition solution. The ratio amplitude of . *ZA*<sup>1</sup> and . *ZA*<sup>2</sup> is . *IA*+ . *<sup>λ</sup>a*∠−30◦ <sup>√</sup><sup>3</sup> . *IA*<sup>+</sup> . *λa* <sup>&</sup>lt; *IA*<sup>+</sup> √3 <sup>3</sup> *λ<sup>a</sup> IA*+*λ<sup>a</sup>* < 1, which means that . *ZA*<sup>1</sup> <sup>&</sup>lt; .

*ZA*2; in other words, the change of phase A in <sup>1</sup> is greater than that of phase A in <sup>2</sup> . Similarly, the change of phase A in <sup>1</sup> is greater than that of phase B in <sup>2</sup> .

It can be concluded from the above that the group of results with the largest variation is the correct solution when single-phase parameters change. In the case of two-phase change, the two parameters are the same, and the normal phase also can be seen as a change phase. It follows that the group of results with the largest variation is also the correct solution. In the case of the normal state or a three-phase fault, the three groups of results are the same, and all are correct solutions. As a result, the correct assumption conditions can be determined by finding the maximum change in the capacitance value, and then the correct solution of the three-phase capacitance value can be calculated. .

It could calculate max *i* = *A*, *B*, *C j* = 1, 2, 3 ( *Zij* <sup>−</sup> . *<sup>Z</sup>*0), where . *Z*<sup>0</sup> is the initial value of single-phase

impedance, and find the max impedance change. Then, this group ( . *ZAj*, . *ZBj* . *ZCj*) is determined as correct calculation. If the parameters of the series reactor are defined as L, and the system frequency is f, then the results of the three-phase capacitance values *CA*,*CB* and *CC* are as follows:

$$\begin{cases} \mathbb{C}\_{A} = \frac{1}{4\pi^{2}f^{2}L - 2\pi f \text{Im}(\bar{Z}\_{Aj})}\\ \mathbb{C}\_{B} = \frac{1}{4\pi^{2}f^{2}L - 2\pi f \text{Im}(\bar{Z}\_{Bj})}\\ \mathbb{C}\_{\mathbb{C}} = \frac{1}{4\pi^{2}f^{2}L - 2\pi f \text{Im}(\bar{Z}\_{\mathbb{C}j})} \end{cases} \tag{7}$$

#### *2.2. Monitoring Criteria*

After solving for the capacitance value, taking into account the normal fluctuations and calculation errors, and in accordance with IEEE Guide for the Protection of Shunt Capacitor Banks [24], the shunt capacitor is considered faulty when the calculated capacitance value

*Ck* and the rated value *CN* do not meet |*Ck* − *CN*| × 100% > 5%. Repair is required. Cst is the statistical data based on the stored capacitance from the protection device. When there is a significant difference between the current capacitance value and the past sample (*CA* ∈/ *Cst*), an alarm is issued. Because relay protection is sensitive to capacitor external fault, and a bus fault would make the voltage zero, the shunt capacitor bank monitor should not trigger an alarm due to an external fault or bus fault. Hence, in order to avoid a monitoring malfunction, a low voltage criterion is added. When a capacitor external fault or bus fault occurs, the voltage will drop significantly. As a result, when any voltage of the three-phase bus is *Uk* < 0.85*UN*, this phase is determined to be a short-circuit fault, the protection will trip and capacitor monitoring does not need an alarm. Considering the need to prevent disturbance, a certain delay is added. The monitoring logic is shown in Figure 4.

**Figure 4.** Monitoring logic.

Considering that capacitor values are affected by environmental factors such as operating temperature, air pressure, dust, etc., adding a year-on-year comparison of capacitance values can reduce the influence of operating conditions on measured values and indicate whether the capacitor has changed significantly after long-term operation; doing so can improve the sensitivity and accuracy of monitoring.

The normal state data for the current and previous years are stored, two sets of capacitance values from the same month are sampled and a paired-sample t test is performed to check for significant differences; if there is a significant difference, the operating state is considered abnormal and an alarm is issued.

First, this assumes that the mean capacitance values of the two months are the same, i.e., there is no significant difference.

Second, the formula for calculating t is given below. *X*1/*X*<sup>2</sup> are the sample data from the previous and current year for the same month. *X*1/*X*<sup>2</sup> are the averages of the two samples. *δX*1/*δX*<sup>2</sup> are the variances of the two samples. *γ* is the correlation coefficient of the two samples.

$$t = \frac{\overline{X\_1} - \overline{X\_2}}{\sqrt{\frac{\delta\_{X\_1}^2 + \delta\_{X\_2}^2 - 2\gamma\delta\_{X\_1}\delta\_{X\_2}}{n-1}}} \tag{8}$$

Third, we assume a confidence level of 95% and *t*(29)0.05 = 2.045 according to the T value table. If *t* ≤ 2.045, no significant difference is found, and the capacitor is operating normally. If *t* > 2.045, a significant difference is found, and the capacitor is operating abnormally.

#### **3. Analysis of Test Results**

To verify the practicability of the monitoring method in this paper, the system is simulated and analyzed by PSCAD/EMTDC simulation software. Figure 5 shows the simulated circuit diagram.

**Figure 5.** Shunt capacitor bank at 10 kV bus.

The simulation system is a shunt capacitor bank built on a 10 kV bus. C0 is the initial capacitance and CX is the variable capacitance, i.e., the simulated capacitance change in the abnormal state. The capacitor branch resistance is 10 Ω, the capacitance value is 9.76 μF and the series reactor is 63.94 mH, given 5% of capacitance.

#### *3.1. Capacitor Internal Fault*

Figure 6 shows the calculation results when phase A capacitance declined by 10% of the standard value. The bus voltage remained unchanged. Phase A RMS current varied from 20.26 A to 18.96 A. Phase B/C RMS current varied from 20.26 A to 19.94 A. As can be seen from the figure, the capacitance changes slightly, the power system still operates normally and the voltage and current do not change significantly. It is correctly identified that the variability of phase A capacitance is out of specification and a phase A alarm is issued. The calculation result of the capacitance is 8.7919 μF; the actual capacitance is 8.784 μF (with less than 0.1% relative error).

**Figure 6.** Phase A internal fault.

Figure 7 shows the simulation when the phase A and B capacitances declined by 10% of the standard capacitance value. The bus voltage remained unchanged. Phase A/B RMS current varied from 20.26 A to 18.55 A. Phase B/C RMS current varied from 20.26 A to 19.54 A. As can be seen from the figure, this method can correctly detect that the capacitance changes of phase A and phase B are out of specification and issue an alarm. The calculated capacitance value for phase A/B is 8.7972 μF; the actual capacitance is 8.784 μF (with only 0.15% relative error).

**Figure 7.** Phase A and B internal fault.

Figure 8 shows the simulation when the capacitances of three phases change at the same time. (A, B and C declined 2%, 4% and 8%, respectively, compared with the standard value). The bus voltage remained unchanged. The phase A RMS current varied from 20.26 A to 19.57 A. The phase B RMS current varied from 20.26 A to 19.37 A. The phase C RMS current varied 20.26 A to 18.97 A. It can be seen from the figure that when there is a deviation in the initial parameters, capacitor monitoring can correctly detect that the phase C capacitance change exceeds specified values and a phase C alarm is triggered. The calculated result of the capacitances of phase A and B are 9.4608 μF (the actual capacitance values are 9.5648 μF and 9.3696 μF, the relative error is 1.09% and 0.97%). The calculated result of phase C capacitance is 8.9873 μF, the actual capacitance is 8.9792 μF and the relative error is 0.09%. There is a certain error in the calculated results of the normal phases A and B due to the initial deviation, but within the acceptable range, the capacitance value of the faulty phase C is still accurately calculated, the abnormality is correctly identified, and an alarm is issued.

**Figure 8.** Phase C internal fault.

The above results show that when some capacitor units are faulty, the slight change of capacitance value can be accurately detected. When the capacitance value changes beyond the shunt capacitor guidelines, or a statistical data comparison flags the value as abnormal, an alarm will be issued, which is convenient for maintenance.

#### *3.2. External Fault*

External faults occur when the fault accrues on a bus or line. The relay protections operate in these cases. The bus voltage almost drops to zero, and the capacitance value calculation is meaningless in this situation. In addition, capacitor monitoring should not trigger an alarm at external fault. Figure 9 shows the normal operation of the bus in the case of a single-phase ground fault. It can be seen from the figure that the capacitance value before and after the fault has a bump, because when calculating the current and voltage, a short circuit in one cycle causes a sudden change during the fault. The capacitance value is stable, and its calculated value is 208.23 μF, because when an external fault occurs, the measured impedance is the impedance from the measuring point to the short circuit point. The external fault should be tripped by the relay protection, and the capacitor monitoring should not trigger an alarm. Adding a low-voltage block make the alarm would not malfunction during the external fault.

**Figure 9.** Phase A ground fault simulation results.

Figure 10 shows the normal operation of the bus in the case of phase A and B grounding faults. During the fault, the calculated phase A and phase B capacitance values are both 208.23 μF; A, B two-phase low-voltage criteria are activated; two-phase capacitor failure monitoring is no longer active; and no malfunction occurred.

**Figure 10.** A and B grounded external fault.

The above results show that the capacitor fault alarm does not malfunction, and the fault is removed by the protection mechanism.

Summarizing various faults of capacitors as shown in Appendix A, it can be seen that this method can correctly detect the internal faults and locate the abnormal phase. This is convenient for maintenance and repair work.

#### **4. Conclusions**

The on-line monitoring method for shunt capacitors proposed in this paper has the following characteristics:

(1) This monitoring method is applicable to shunt capacitor banks of all connection types. It can be realized only by using a relay protection device, with no additional device to measure the state quantity. The method is economical and convenient.

(2) In the event of a slight capacitor failure or abnormality, the abnormal phase can be detected to maintain safe operation.

**Author Contributions:** Conceptualization, Y.L. and Z.W.; methodology, Y.L.; software, Y.L.; validation, J.G.; formal analysis, Z.W.; investigation, J.G.; resources, J.G.; data curation, J.G.; writing original draft preparation, Y.L.; writing—review and editing, Y.L.; visualization, J.G.; supervision, Z.W.; project administration, J.G.; funding acquisition, J.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data presented in this study are openly available in IEEE, at references [13,24].

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**


**Table A1.** Alarm results in each capacitor fault.

#### **References**


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