Impact of Inter-Turn Short Circuit in Excitation Windings on Magnetic Field and Stator Current of Synchronous Condenser under Unbalanced Voltage

: Inter-turn short circuit in the excitation windings of synchronous condensers is a common fault that directly impacts their normal operation. However, current fault analysis and diagnosis of synchronous condensers primarily rely on voltage-balanced conditions, while research on short-circuit faults under unbalanced voltage conditions is limited. Therefore, this paper aims to analyze the fault characteristics of inter-turn short circuits in the excitation windings of synchronous condensers under unbalanced grid voltage. Mathematical models were developed to represent the air gap ﬂux density and stator parallel currents for four operating conditions: normal operation and inter-turn short circuit fault under balanced voltage, as well as a process without fault and with inter-turn short circuit fault under unbalanced voltage. By comparing the harmonic content and amplitudes, various aspects of the fault mechanism of synchronous condensers were revealed, and the operating characteristics under different conditions were analyzed. Considering the four aforementioned operating conditions, ﬁnite element simulation models were created for the TTS-300-2 synchronous condenser in a speciﬁc substation as a case study. The results demonstrate that the inter-turn short circuit fault in the excitation windings under unbalanced voltage leads to an increase in even harmonic currents in the stator parallel currents, particularly the second and fourth harmonics. This validates the accuracy of the theoretical analysis ﬁndings.


Introduction
Currently, the ultra-high voltage direct current (UHVDC) system is rapidly developing, imposing more significant requirements on reactive power within the power grid.Largescale synchronous condensers (LSSC), with high capacity, exhibit exceptional transient reactive power support and short-term overload capabilities.By positioning LSSC at the inverter end of a weak AC grid, commutation failures can be effectively prevented, and fault clearing speed can be accelerated [1].Consequently, ensuring the reliable operation of LSSC is crucial in enhancing the dynamic voltage stability of power systems and ensuring the stable operation of UHVDC transmission [2].
Inter-turn short circuit in the excitation windings of synchronous condensers is a common fault not only limits the reactive power capability but also increases the excitation current, power losses, and local temperature of the synchronous condenser.In severe cases, it can exacerbate rotor vibrations, generate significant axial magnetization, and even completely shut down the synchronous condenser [3].Analyzing characteristic patterns and accurately diagnosing inter-turn short circuit faults in the excitation windings are complex tasks within the field of system engineering.Fault diagnosis is typically carried out using methods such as the DC resistance method [4], AC impedance method [5], and repetitive pulse method [6].Since synchronous condensers share structural similarities Energies 2023, 16, 5695 2 of 18 with synchronous generators, research findings related to synchronous generators can be utilized for synchronous condensers [7][8][9][10][11].M. Xu et al. [12] comprehensively analyzed the stator circulating current within parallel branches, considering various degrees and positions of rotor inter-turn short circuits.The analysis was performed using the finite element method, yielding valuable insights into the behavior of the circulating current.Currently, several studies have been conducted on inter-turn short circuit faults in the excitation windings of synchronous condensers.G. Xu et al. [13] present the single-phase short circuit faults' electromagnetic and temperature field calculation and the experimental validation.M. Ma et al. [14] investigated inter-turn short circuit faults in the rotor windings of synchronous condensers by analyzing commutation failures in high-voltage direct current transmission.Finite element analysis revealed that commutation failures can cause abnormal vibrations in the rotor side of the synchronous condenser, which positively correlates with the severity of the fault.Y. Zhang et al. [15] applied wavelet transform to preprocessed excitation current signals extracted from normal and faulty states of the synchronous condenser.The extracted features were input into a radial basis function neural network for fault diagnosis.Z. Chen et al. [16] analyzed inter-turn short circuit faults in the rotor windings of synchronous condensers from the perspective of temperature distribution and validated the analysis through finite element simulations.C. Wei et al. [17] investigated the relationship between the number of short-circuited turns in the excitation windings of synchronous condensers and the magnetic field current in the rotor windings, proposing an online monitoring fault diagnosis strategy.A. N. Novozhilov et al. [18] established a mathematical model for inter-turn short circuit faults in the excitation windings of synchronous condensers, achieving an accuracy range of 5% to 10%.Most existing research on the mentioned faults primarily focuses on the short circuit between turns in the excitation winding.However, the voltage waveform of the grid deviates from a standard sinusoidal shape due to the non-standard sinusoidal waveform generated by most generators.As a result, when a short circuit occurs in the excitation winding, it can be seen as a combination of unbalanced voltage and a short circuit between turns in the excitation winding.The prevalence of unbalanced voltage further complicates the fault analysis, as the fault environments studied in literature may not accurately reflect real-world scenarios.Consequently, matching the observed fault characteristics of the synchronous condenser with known fault patterns during on-site diagnostics becomes challenging, often leading to incorrect or even misdiagnosis.Therefore, it is crucial to research the compound fault of unbalanced voltage and short circuits in the excitation winding.This research aims to identify distinctive fault characteristics specific to this type of fault and differentiate them from fault characteristics caused by individual faults.Such investigations are essential for achieving accurate diagnosis and establishing reliable diagnostic criteria in scenarios involving these compound faults.J. LI et al. [19] conducted a simulation study on the inter-turn short circuit fault in the stator winding of a doubly-fed induction generator.They used finite element modeling to analyze the changes in stator line voltage, rotor line current, and electromagnetic torque when the excitation winding experiences an inter-turn short circuit fault, considering the presence of inherent grid voltage imbalance and static eccentricity.However, their study [19] focused solely on simulation modeling analysis and did not investigate the theoretical research on mathematical expressions of the relevant fault characteristic quantities after the occurrence of the fault.As a result, their study has certain limitations that need to be addressed.
Among the electrical characteristic-oriented methods, the current-based method is most widely employed since it does not require extra equipment and can make full use of the stator winding as search coils for further processing.In addition, the parallel branch circulating current signal in the stator winding carries valuable fault information and, in certain cases, offers more effective fault diagnosis than rotor vibration signals.To address the issue of voltage imbalance, this paper begins by investigating the air gap flux density and proceeds to derive expressions for both the air gap flux density and stator parallel branch circulating current of the synchronous condenser in four distinct operating conditions.Additionally, it conducts a comprehensive analysis of the characteristics of the stator parallel branch circulating current under different operating conditions, shedding light on their intricate dynamics: normal operation of the synchronous condenser under balanced voltage, inter-turn short circuit fault in the excitation windings of the synchronous condenser under balanced voltage, normal operation of the synchronous condenser under voltage imbalance, and inter-turn short circuit fault in the excitation windings of the synchronous condenser under voltage imbalance.The degree of voltage imbalance is varied by adjusting the voltage magnitude.Mathematical representations are derived, and a finite element simulation model of the synchronous condenser is constructed to validate the analysis.This study aims to provide a theoretical basis for diagnosing interturn short circuit faults in the excitation windings of synchronous condensers under voltage imbalance conditions.
The main structure of this paper is as follows: In Section 2, the expressions for the air gap flux density and stator parallel currents under different fault types of a synchronous condenser are derived, and the impacts of faults are analyzed from a theoretical perspective.Section 3 validates the proposed analytical model's effectiveness and accuracy using finite element analysis, and the changes in air gap flux density and stator parallel currents after faults in the synchronous condenser are analyzed using the finite element model, which is consistent with the theoretical analysis.Finally, Section 4 summarizes the conclusions of this paper.

Analysis of Composite Faults in Synchronous Condensers
To compare the variations in air gap magnetic flux density, stator parallel current amplitude, and harmonic content under different operating conditions, this paper focuses on four scenarios: normal operation of the synchronous condenser under balanced voltage, inter-turn short circuit fault in the excitation windings of the synchronous condenser under balanced voltage, normal operation of the synchronous condenser under voltage imbalance, and inter-turn short circuit fault in the excitation windings of the synchronous condenser under voltage imbalance.Compared to symmetrical grid voltage, it is assumed that when the grid voltage becomes asymmetric, the amplitude of one or two phases is reduced.Nevertheless, the derived formulas apply to all cases, demonstrating their universality.

Air Gap Magnetic Potential during Normal Operation of Synchronous Condenser under Balanced Grid Voltage
Neglecting higher-order harmonics, the air gap magnetic potential of the synchronous condenser during normal operation can be expressed as follows: (1) where ω is the stator current angular frequency and rotor rotational angular velocity, α m is the stator spatial electrical angle, ψ is the internal power factor angle, F s is the armature magnetic flux amplitude, and F r is the excitation magnetic flux amplitude.

Air Gap Magnetic Potential during Inter-Turn Short Circuit Fault in the Excitation Windings of Synchronous Condenser under Balanced Grid Voltage
When an inter-turn short circuit occurs in the excitation windings of the synchronous condenser, it generates a counteracting magnetic field [20].The reverse magnetic MMF (Magneto-Motive Force) generated by the short-circuited winding is given by where I f0 is the excitation current, N short is the number of turns in the short-circuited winding in the same slot, α is the mechanical angle between the slot where the shortcircuited winding is located and the adjacent slot, θ r is the mechanical angle of the rotor.
The magnetic potential resulting from the short circuit can be expressed in terms of its harmonic components through Fourier decomposition: After performing the Fourier transform on F d (θ r ), it can be observed that the main magnetic field in the air gap exhibits various harmonics.When α = 2 kπ/n, taking n = 1,2, and θ r = ωt − α m , we have

The Air Gap Magnetic Potential of the Synchronous Condenser under Unbalanced Grid Voltage Conditions without Any Faults Occurring
Under balanced grid voltage, the armature magnetic field of the synchronous condenser exhibits a synchronized circular rotation with the rotor.However, when the grid voltage becomes unbalanced, the armature magnetic field undergoes distortion, assuming an elliptical shape.The symmetrical component method can be applied to characterize the armature magnetic field expression in such scenarios.In this method, the positive-sequence armature magnetic field rotates synchronously with the rotor, while the negative-sequence armature magnetic field rotates in the opposite direction.Meanwhile, the zero-sequence armature magnetic field remains at zero [21].Consequently, in the presence of an unbalanced grid voltage, the expression for the armature magnetic potential is given by where F s + is the positive-sequence armature magnetic flux amplitude, F s − is the negativesequence armature magnetic flux amplitude.Based on the previous assumption, both F s + and F s − are smaller than F s .The negative-sequence magnetic field caused by the unbal- anced grid voltage induces a double-frequency current in the rotor winding.Therefore, the expression for the excitation current is as follows: where I f0 is the direct current excitation current generated by the generator excitation system, I f2 is the amplitude of the twice-frequency excitation current induced.The expression for the excitation magnetic field generated by the excitation current is as follows: where k is the waveform coefficient of the excitation magnetic flux.Therefore, the expression for the synthesized air-gap magnetic potential generated under unbalanced voltage conditions is as follows: Equations ( 14)- (18) reveal that unbalanced voltage conditions lead to the induction of a double-frequency current in the excitation winding of the synchronous condenser.As a consequence, third harmonic components are produced in the air gap.It is worth noting that the amplitude of the third harmonic exhibits a direct proportionality to the double-frequency current.

The Air Gap Magnetic Flux in the Synchronous Condenser When There Is an
Inter-Turn Short Circuit in the Excitation Winding under Unbalanced Grid Voltage Under unbalanced grid voltage, a negative-sequence magnetic field is produced, generating double-frequency excitation current in the rotor.Therefore, it becomes essential to account for the influence of this double-frequency current on the excitation magnetic flux when investigating inter-turn short circuits in the excitation winding.The expression for the air gap magnetic flux in the synchronous condenser under these circumstances is as follows: Energies 2023, 16, 5695 6 of 18 According to Equations ( 19)-( 25), the presence of unbalanced voltage, coupled with an inter-turn short circuit in the excitation winding, gives rise to a multifaceted air gap magnetic flux in the synchronous condenser.This flux encompasses several components, namely even harmonic components induced by the inter-turn short circuit, a DC component introduced by the rotor's double-frequency current, and odd harmonic components.

Air Gap Magnetic Flux Density
During operation, the air gap magnetic flux density of the synchronous condenser, denoted as Λ 0 per unit area, remains constant.The following expression is derived to represent the air gap magnetic flux density:

Analysis of Parallel Branch Current Circulation in the Stator
The stator winding of a large synchronous condenser is connected in a double Y configuration, employing a three-phase double-layer winding form.This configuration comprises two parallel branches for each phase (A, B, and C), with multiple coil windings connected in series within each branch.Hence, the expression for the instantaneous value of the induced electromotive force in a single branch of the parallel stator winding of the synchronous condenser is as follows: where l is the air gap length, f is the electrical frequency, N z is the number of conductors connected in series in a single stator branch, and k w1 is the fundamental winding factor.Figure 1 illustrates the equivalent circuit of the parallel branch in phase A of the synchronous condenser.The circuit includes R a1 , R a2 , X a1 , and X a2 , which represent the resistance and leakage reactance of the two branches.The circulating current is denoted as i c .
Energies 2023, 16, 5695 where l is the air gap length, f is the electrical frequency, Nz is the number of conductors connected in series in a single stator branch, and kw1 is the fundamental winding factor Figure 1 illustrates the equivalent circuit of the parallel branch in phase A of the synchro nous condenser.The circuit includes Ra1, Ra2, Xa1, and Xa2, which represent the resistance and leakage reactance of the two branches.The circulating current is denoted as ic.When the synchronous condenser is operating normally, applying Kirchhoff's Volt age Law (KVL) to the stator parallel circuit yields the following equation: The expression for the stator parallel circuit current can be obtained from Equation (28) as follows:

Under the Condition of Balanced Grid Voltage, the Synchronous Condenser Operates without Any Faults
The induced electromotive force in the two parallel branches during the normal op eration of the synchronous condenser is given by the following expression: When the synchronous condenser is operating normally, applying Kirchhoff's Voltage Law (KVL) to the stator parallel circuit yields the following equation: The expression for the stator parallel circuit current can be obtained from Equation (28) as follows:

Under the Condition of Balanced Grid Voltage, the Synchronous Condenser Operates without Any Faults
The induced electromotive force in the two parallel branches during the normal operation of the synchronous condenser is given by the following expression: The stator parallel branch current in this case is Based on Equation (31), it can be concluded that during normal operation of the synchronous condenser, the stator parallel branch current is zero, indicating the absence of any current flowing through the stator parallel branches in this case.

Grid Voltage Balance Synchronous Condenser Excitation Winding Inter-Turn Short Circuit Occurs
When a short circuit occurs in the excitation winding of the synchronous condenser, the induced electromotive force in the two parallel branches can be expressed as follows: The parallel stator current is given by Based on the above analysis, it can be concluded that during an inter-turn short circuit in the excitation winding of the synchronous condenser under balanced grid voltage, the stator parallel current exhibits a second harmonic component that is directly proportional to F d2 .In other words, the magnitude of the parallel current is directly related to the severity of the short circuit.

The Air Gap Magnetic Potential of the Synchronous Condenser under Unbalanced Grid Voltage Conditions without Any Faults Occurring
In the case of unbalanced grid voltage, the induced electromotive force in the two parallel branches of the stator is given by the following expression: At this time, the stator parallel current is given by Based on the analysis, it can be concluded that during the operation of the synchronous condenser under unbalanced grid voltage conditions, there are no stator parallel currents induced under normal operation.

The Air Gap Magnetic Flux in the Synchronous Condenser When There Is an Inter-Turn Short Circuit in the Excitation Winding under Unbalanced Grid Voltage
In the presence of an inter-turn short circuit in the excitation winding of the synchronous condenser under unbalanced grid voltage, the induced electromotive forces in the two stator branches can be expressed as follows: At this time, the parallel stator current is given by Based on the theoretical analysis presented, it can be concluded that the parallel stator current in the synchronous condenser, under the combined conditions of unbalanced grid voltage and inter-turn short circuit in the excitation winding, is predominantly characterized by even harmonics.The magnitude of the parallel stator current is determined by the degree of grid voltage unbalance and the severity of the inter-turn short circuit.
Inter-turn short circuit faults in the excitation winding of the synchronous condenser result in significant modifications to the air gap flux density and parallel stator current.Unbalanced grid voltage conditions further influence these changes in characteristics.Consequently, these distinctive variations can serve as reliable references for fault diagnosis.A comprehensive verification will be conducted through finite element simulation to validate our findings.

Simulation Analysis
This paper presents a case study on the TTS-300-2 type novel synchronous condenser at a specific power station.To investigate its behavior, a two-dimensional (2D) finite element simulation model and its corresponding external circuit are developed using Ansys Maxwell 2021 R1 software.This approach enables a comprehensive analysis of the synchronous condensers' performance and facilitates valuable insights into its operation.The utilization of adaptive meshing in condensor modeling offers significant advantages by automatically adjusting the grid density in response to electromagnetic field variations.This adaptive approach enhances the accuracy of simulation results, improves computational efficiency, and optimizes both time and computational resources.In this paper, we employed an adaptive mesh design for the simulation.Details regarding the mesh partition can be found in Table 1.The Finite Element Method is a numerical technique used to solve integral and partial differential equations, offering superior accuracy compared to other analytical analyses.It employs magnetic linearized parameters to accurately model electromagnetic phenomena.In this study, a 2D field-circuit coupled model of the synchronous condenser is developed in ANSYS Maxwell using the Finite Element Method.It is important to note that, for simplicity, the model neglects the effects of skin effects and eddy current losses.The electromagnetic field expression for the electrical machine is represented by Equation (38).
where A is the axial components of the magnetic vector potential; A 0 is the magnetic vector potential in the first boundary; J Z is source current density; µ is material reluctivity.2. In Figure 3, the symbols LPhaseA, LPhaseA1, LPhaseB, LPhaseB1, LPhaseC, and LPhaseC1 represent the windings of the three-phase double parallel branches.The excitation winding is denoted as LField, while the faulty portion responsible for the inter-turn short circuit is referred to as LShortWinding.By manipulating the parameters LField and LShortWinding, precise control over the number of turns in the short-circuited winding can be achieved.Furthermore, the resistances LR and LShortR correspond to LField and LShortWinding, respectively, and must be adjusted accordingly when altering the number of turns.

Model Accuracy Verification
Given the short operating time and the absence of actual on-site fault data for large synchronous condensers, conducting direct short-circuit experiments on-site is impractical.Therefore, to validate the accuracy of the simulation model, simulations were performed under rated operating conditions to analyze the output torque, stator phase voltage, and phase current on the rotor shaft.The rated operating condition of the synchronous condenser refers to its operation at the rated voltage provided in Table 2, carrying the rated load and being connected to the power grid.The specific results are illustrated in Figures 4-6.Additionally, we conducted a comparative analysis between the actual data of stator phase voltage and stator phase current obtained during the synchronous condenser's rated operation and the corresponding simulation data.The detailed results are presented in Table 3. From the figures, it is evident that the average output torque of the synchronous condenser is zero, which aligns with its expected operational state.Moreover, the phase voltages and currents comply with the rated parameters, and the error falls within an acceptable range.The three-phase currents exhibit symmetry, and there exists a time gap of approximately 5 ms between the stator phase current and the phase voltage.This gap indicates that the stator phase current leads the phase voltage by 90°.These observations confirm the accuracy of the simulation model.By accurately simulating the output torque, stator phase voltage, and phase current under rated conditions, the simulation model has demonstrated its ability to replicate the behavior of the synchronous condenser.Despite the challenges posed by the lack of actual fault data and on-site experiments, the validated simulation results instill confidence in the reliability of the model for further analysis and fault diagnosis.
The 2D finite element simulation model of the TTS-300-2 synchronous condenser used in this paper was based on operational parameters primarily extracted from the device's technical manual.Key electrical and structural parameters from Table 2 were

Model Accuracy Verification
Given the short operating time and the absence of actual on-site fault data for large synchronous condensers, conducting direct short-circuit experiments on-site is impractical.Therefore, to validate the accuracy of the simulation model, simulations were performed under rated operating conditions to analyze the output torque, stator phase voltage, and phase current on the rotor shaft.The rated operating condition of the synchronous condenser refers to its operation at the rated voltage provided in Table 2, carrying the rated load and being connected to the power grid.The specific results are illustrated in Figures 4-6.Additionally, we conducted a comparative analysis between the actual data of stator phase voltage and stator phase current obtained during the synchronous condenser's rated operation and the corresponding simulation data.The detailed results are presented in Table 3. From the figures, it is evident that the average output torque of the synchronous condenser is zero, which aligns with its expected operational state.Moreover, the phase voltages and currents comply with the rated parameters, and the error falls within an acceptable range.The three-phase currents exhibit symmetry, and there exists a time gap of approximately 5 ms between the stator phase current and the phase voltage.This gap indicates that the stator phase current leads the phase voltage by 90°.These observations confirm the accuracy of the simulation model.By accurately simulating the output torque, stator phase voltage, and phase current under rated conditions, the simulation model has demonstrated its ability to replicate the behavior of the synchronous condenser.Despite the challenges posed by the lack of actual fault data and on-site experiments, the validated simulation results instill confidence in the reliability of the model for further analysis and fault diagnosis.
The 2D finite element simulation model of the TTS-300-2 synchronous condenser used in this paper was based on operational parameters primarily extracted from the device's technical manual.Key electrical and structural parameters from Table 2 were

Model Accuracy Verification
Given the short operating time and the absence of actual on-site fault data for large synchronous condensers, conducting direct short-circuit experiments on-site is impractical.Therefore, to validate the accuracy of the simulation model, simulations were performed under rated operating conditions to analyze the output torque, stator phase voltage, and phase current on the rotor shaft.The rated operating condition of the synchronous condenser refers to its operation at the rated voltage provided in Table 2, carrying the rated load and being connected to the power grid.The specific results are illustrated in Figures 4-6.Additionally, we conducted a comparative analysis between the actual data of stator phase voltage and stator phase current obtained during the synchronous condenser's rated operation and the corresponding simulation data.The detailed results are presented in Table 3.
From the figures, it is evident that the average output torque of the synchronous condenser is zero, which aligns with its expected operational state.Moreover, the phase voltages and currents comply with the rated parameters, and the error falls within an acceptable range.The three-phase currents exhibit symmetry, and there exists a time gap of approximately 5 ms between the stator phase current and the phase voltage.This gap indicates that the stator phase current leads the phase voltage by 90 • .These observations confirm the accuracy of the simulation model.By accurately simulating the output torque, stator phase voltage, and phase current under rated conditions, the simulation model has demonstrated its ability to replicate the behavior of the synchronous condenser.Despite the challenges posed by the lack of actual fault data and on-site experiments, the validated simulation results instill confidence in the reliability of the model for further analysis and fault diagnosis.accurately applied during the model setup to ensure alignment with the equipment's characteristics.Furthermore, by conducting simulations at the rated voltage and load, we validated the accuracy of torque, phase voltage, and phase current output results under the TTS-300-2 synchronous condenser's rated operating conditions.As a result, the selected parameters effectively mimic the actual device's operational characteristics, guaranteeing the reliability of the simulation analysis.accurately applied during the model setup to ensure alignment with the equipment's characteristics.Furthermore, by conducting simulations at the rated voltage and load, we validated the accuracy of torque, phase voltage, and phase current output results under the TTS-300-2 synchronous condenser's rated operating conditions.As a result, the selected parameters effectively mimic the actual device's operational characteristics, guaranteeing the reliability of the simulation analysis.accurately applied during the model setup to ensure alignment with the equipment's characteristics.Furthermore, by conducting simulations at the rated voltage and load, we validated the accuracy of torque, phase voltage, and phase current output results under the TTS-300-2 synchronous condenser's rated operating conditions.As a result, the selected parameters effectively mimic the actual device's operational characteristics, guaranteeing the reliability of the simulation analysis.The 2D finite element simulation model of the TTS-300-2 synchronous condenser used in this paper was based on operational parameters primarily extracted from the device's technical manual.Key electrical and structural parameters from Table 2 were accurately applied during the model setup to ensure alignment with the equipment's characteristics.Furthermore, by conducting simulations at the rated voltage and load, we validated the accuracy of torque, phase voltage, and phase current output results under the TTS-300-2 synchronous condenser's rated operating conditions.As a result, the selected parameters effectively mimic the actual device's operational characteristics, guaranteeing the reliability of the simulation analysis.
To simulate the short-circuit of the excitation winding, this study conducted simulation analyses with different numbers of coil turns (1 turn, 3 turns, and 5 turns) short-circuited in the first slot near the large tooth side.In the voltage imbalance simulation, the B-phase and C-phase voltages were maintained at their rated values.In contrast, only the magnitude of the A-phase voltage was adjusted to control the degree of voltage imbalance in the power grid.Specifically, the A-phase voltage values of 96%, 93%, and 90% of the rated voltage were chosen to highlight the simulation results.Due to space constraints, only a partial waveform is presented in this section.

•
Analysis of Air Gap Magnetic Flux Density Figure 7a,b depict the analysis results of air gap magnetic flux density under balanced grid voltage with a short circuit fault.Analysis of Figure 7a reveals that a rotor inter-turn short circuit leads to a reduction in air gap magnetic flux density due to a loss of magnetic potential.The severity of the inter-turn short circuit corresponds to a more pronounced decay in the magnetic flux density.Figure 7b demonstrates that in the absence of an interturn short circuit, the air gap magnetic flux density is primarily composed of fundamental frequency and odd harmonic components, with only minor influence from slotting effects in the stator and rotor, resulting in a small amount of even harmonic components.The occurrence of a second harmonic in the air gap magnetic density under normal conditions could be attributed to simulation errors resulting in non-uniformity within the air gap magnetic field.However, the presence of an inter-turn short circuit significantly increases the even harmonic magnetic flux density, which intensifies as the severity of the short circuit increases.This observation confirms the accuracy of the derived magnetic field theory.

• Analysis of Parallel Circulation between Stator Branches
Figure 8a,b depict the analysis results of stator parallel branch circulating currents under balanced grid voltage with a short circuit fault.In normal operating conditions, no circulating current or harmonic component is present in the stator parallel branches.However, the occurrence of a short circuit in the excitation winding leads to the generation of circulating currents in the parallel branches, with a predominant presence of even harmonic circulating currents, especially the second harmonic component.Moreover, as the fault severity increases, the amplitude of the harmonic circulating currents also intensifies.This observation confirms the accuracy of the theoretical derivation.
slotting effects in the stator and rotor, resulting in a small amount of even harmonic components.The occurrence of a second harmonic in the air gap magnetic density under normal conditions could be attributed to simulation errors resulting in non-uniformity within the air gap magnetic field.However, the presence of an inter-turn short circuit significantly increases the even harmonic magnetic flux density, which intensifies as the severity of the short circuit increases.This observation confirms the accuracy of the derived magnetic field theory.•

Analysis of Parallel Circulation between Stator Branches
Figure 8a,b depict the analysis results of stator parallel branch circulating currents under balanced grid voltage with a short circuit fault.In normal operating conditions, no circulating current or harmonic component is present in the stator parallel branches.However, the occurrence of a short circuit in the excitation winding leads to the generation of

Analysis of Air Gap Flux Density and Stator Parallel Circulation When the Grid
Voltage Is Unbalanced and the Synchronous Condenser Is Fault-Free

Analysis of Air Gap Magnetic Flux Density
Figure 9a,b showcase the analysis results of air gap flux density under the condition of unbalanced grid voltage.Based on the preceding analysis, it is evident that unbalanced grid voltage causes a reduction in air gap flux density, with a greater degree of voltage imbalance resulting in a more substantial loss of air gap flux density.In the presence of voltage imbalance, the amplitude of the fundamental component exhibits a negative correlation with the degree of voltage imbalance.In contrast, the amplitude of the third harmonic component demonstrates a positive correlation.This behavior can be attributed to the induction of twice the frequency current in the excitation winding by the negativesequence magnetic fields caused by voltage imbalance.Consequently, third-harmonic magnetic fields are superimposed on the existing third harmonic, leading to an amplified amplitude of the third harmonic and an increased level of magnetic field distortion.According to Equation ( 14), voltage imbalance in the grid leads to the emergence of third

Analysis of Air Gap Flux Density and Stator Parallel Circulation When the Grid
Voltage Is Unbalanced and the Synchronous Condenser Is Fault-Free

•
Analysis of Air Gap Magnetic Flux Density Figure 9a,b showcase the analysis results of air gap flux density under the condition of unbalanced grid voltage.Based on the preceding analysis, it is evident that unbalanced grid voltage causes a reduction in air gap flux density, with a greater degree of voltage imbalance resulting in a more substantial loss of air gap flux density.In the presence of voltage imbalance, the amplitude of the fundamental component exhibits a negative correlation with the degree of voltage imbalance.In contrast, the amplitude of the third harmonic component demonstrates a positive correlation.This behavior can be attributed to the induction of twice the frequency current in the excitation winding by the negative-sequence magnetic fields caused by voltage imbalance.Consequently, third-harmonic magnetic fields are superimposed on the existing third harmonic, leading to an amplified amplitude of the third harmonic and an increased level of magnetic field distortion.According to Equation ( 14), voltage imbalance in the grid leads to the emergence of third harmonic components in the air gap magnetic field.Consequently, the magnetic field assumes an elliptical shape, deviating from the circular shape predicted by Equation (1).This elliptical magnetic field causes variations in both the amplitude and harmonic content of the air gap magnetic field at different time points, as illustrated in Table 4. Table 4 illustrates the temporal variations of the magnetic field under unbalanced voltage conditions.The primary changes observed are in the amplitude and fundamental amplitude.Although the third harmonic also undergoes changes, its magnitude remains relatively small and inconspicuous due to the low degree of failure.10 depicts the analysis results of the stator parallel branch current under the condition of unbalanced grid voltage.The theoretical analysis, as indicated by Equations (34) and (35), suggests that the stator parallel branch current remains unaffected by unbalanced grid voltage and maintains a value of 0. However, in the simulation, various factors, such as magnetic saturation and slot effects that occur during the operation of the synchronous condenser, are considered, resulting in a negligible non-zero value for the stator parallel branch current.Although present, the magnitude of this current is minimal and can be disregarded.Thus, during normal operation without faults, the synchronous condenser does not generate any significant stator parallel branch current in unbalanced grid voltage.

• Analysis of Parallel Circulation between Stator Branches
Figure 10 depicts the analysis results of the stator parallel branch current under the condition of unbalanced grid voltage.The theoretical analysis, as indicated by Equations (34) and (35), suggests that the stator parallel branch current remains unaffected by unbalanced grid voltage and maintains a value of 0. However, in the simulation, various factors, such as magnetic saturation and slot effects that occur during the operation of the synchronous condenser, are considered, resulting in a negligible non-zero value for the stator parallel branch current.Although present, the magnitude of this current is minimal and can be disregarded.Thus, during normal operation without faults, the synchronous condenser does not generate any significant stator parallel branch current in unbalanced grid voltage.12a,b depict the analysis results of air gap magnetic flux density under varying levels of grid voltage imbalance with excitation winding turn-to-turn short circuits.The observations from these figures reveal that unbalanced grid voltage introduces distortions in the air gap magnetic flux density, which are further amplified in the presence of turn-to-turn short circuit faults in the excitation winding.Consequently, there is a significant loss of air gap magnetic flux density.Furthermore, the combined occurrence of faults exacerbates the loss of air gap magnetic flux density compared to a single fault scenario.As the degree of turn-to-turn short circuit in the excitation winding intensifies, even harmonics, particularly the second harmonic, become more prominent in the air gap magnetic flux density.Conversely, grid voltage imbalance primarily affects the fundamental and third harmonic components of the air gap magnetic flux density.The fundamental magnetic flux density decreases while the third harmonic magnetic flux density increases with increasing voltage imbalance.These factors collectively influence the various harmonic components.The simulation results align with the theoretical analysis conducted.12a,b depict the analysis results of air gap magnetic flux density under varying levels of grid voltage imbalance with excitation winding turn-to-turn short circuits.The observations from these figures reveal that unbalanced grid voltage introduces distortions in the air gap magnetic flux density, which are further amplified in the presence of turn-to-turn short circuit faults in the excitation winding.Consequently, there is a significant loss of air gap magnetic flux density.Furthermore, the combined occurrence of faults exacerbates the loss of air gap magnetic flux density compared to a single fault scenario.As the degree of turn-to-turn short circuit in the excitation winding intensifies, even harmonics, particularly the second harmonic, become more prominent in the air gap magnetic flux density.Conversely, grid voltage imbalance primarily affects the fundamental and third harmonic components of the air gap magnetic flux density.The fundamental magnetic flux density decreases while the third harmonic magnetic flux density increases with increasing voltage imbalance.These factors collectively influence the various harmonic components.The simulation results align with the theoretical analysis conducted.12a,b depict the analysis results of air gap magnetic flux density under varying levels of grid voltage imbalance with excitation winding turn-to-turn short circuits.The observations from these figures reveal that unbalanced grid voltage introduces distortions in the air gap magnetic flux density, which are further amplified in the presence of turn-to-turn short circuit faults in the excitation winding.Consequently, there is a significant loss of air gap magnetic flux density.Furthermore, the combined occurrence of faults exacerbates the loss of air gap magnetic flux density compared to a single fault scenario.As the degree of turn-to-turn short circuit in the excitation winding intensifies, even harmonics, particularly the second harmonic, become more prominent in the air gap magnetic flux density.Conversely, grid voltage imbalance primarily affects the fundamental and third harmonic components of the air gap magnetic flux density.The fundamental magnetic flux density decreases while the third harmonic magnetic flux density increases with increasing voltage imbalance.These factors collectively influence the various harmonic components.The simulation results align with the theoretical analysis conducted.

• Analysis of Parallel Circulation between Stator Branches
Figure 13a,b display the analysis results of stator parallel currents under different levels of the rotor winding inter-turn short circuits and voltage imbalance in the power grid.Likewise, Figure 14a,b illustrate the analysis results of stator parallel currents under varying levels of the rotor winding inter-turn short circuit and voltage balance in the power grid.Building upon the simulation results and the theoretical analysis discussed earlier, it is evident that voltage imbalance in the power grid does not impact stator parallel currents in the absence of a short circuit fault.However, following the occurrence of the rotor winding inter-turn short circuit, voltage imbalance in the power grid influences the waveform of stator parallel currents.Consequently, in the case of compound faults, the severity of rotor winding inter-turn short circuits and the degree of voltage imbalance in the power grid affect stator parallel currents.As the severity of rotor winding inter-turn short circuits and voltage imbalance in the power grid intensifies, even harmonics, particularly the second and fourth harmonics, become more prominent in the stator parallel currents.Nonetheless, compared to the severity of rotor winding inter-turn short circuits, the influence of voltage imbalance on stator parallel currents is relatively minor and more susceptible to environmental interference during measurement.Therefore, fault diagnosis of the excitation winding inter-turn short circuit should consider multiple fault characteristics to ensure accurate assessment.

• Analysis of Parallel Circulation between Stator Branches
Figure 13a,b display the analysis results of stator parallel currents under different levels of the rotor winding inter-turn short circuits and voltage imbalance in the power grid.Likewise, Figure 14a,b illustrate the analysis results of stator parallel currents under varying levels of the rotor winding inter-turn short circuit and voltage balance in the power grid.Building upon the simulation results and the theoretical analysis discussed earlier, it is evident that voltage imbalance in the power grid does not impact stator parallel currents in the absence of a short circuit fault.However, following the occurrence of the rotor winding inter-turn short circuit, voltage imbalance in the power grid influences the waveform of stator parallel currents.Consequently, in the case of compound faults, the severity of rotor winding inter-turn short circuits and the degree of voltage imbalance in the power grid affect stator parallel currents.As the severity of rotor winding inter-turn short circuits and voltage imbalance in the power grid intensifies, even harmonics, particularly the second and fourth harmonics, become more prominent in the stator parallel currents.Nonetheless, compared to the severity of rotor winding inter-turn short circuits, the influence of voltage imbalance on stator parallel currents is relatively minor and more susceptible to environmental interference during measurement.Therefore, fault diagnosis of the excitation winding inter-turn short circuit should consider multiple fault characteristics to ensure accurate assessment.

• Analysis of Parallel Circulation between Stator Branches
Figure 13a,b display the analysis results of stator parallel currents under different levels of the rotor winding inter-turn short circuits and voltage imbalance in the power grid.Likewise, Figure 14a,b illustrate the analysis results of stator parallel currents under varying levels of the rotor winding inter-turn short circuit and voltage balance in the power grid.Building upon the simulation results and the theoretical analysis discussed earlier, it is evident that voltage imbalance in the power grid does not impact stator parallel currents in the absence of a short circuit fault.However, following the occurrence of the rotor winding inter-turn short circuit, voltage imbalance in the power grid influences the waveform of stator parallel currents.Consequently, in the case of compound faults, the severity of rotor winding inter-turn short circuits and the degree of voltage imbalance in the power grid affect stator parallel currents.As the severity of rotor winding inter-turn short circuits and voltage imbalance in the power grid intensifies, even harmonics, particularly the second and fourth harmonics, become more prominent in the stator parallel currents.Nonetheless, compared to the severity of rotor winding inter-turn short circuits, the influence of voltage imbalance on stator parallel currents is relatively minor and more susceptible to environmental interference during measurement.Therefore, fault diagnosis of the excitation winding inter-turn short circuit should consider multiple fault characteristics to ensure accurate assessment.

Conclusions
This paper examines the impact of inter-turn short circuits in the excitation winding and unbalanced grid voltage on the air gap magnetic flux density and circulating current between stator parallel branches in a synchronous condenser.The study presents derived mathematical expressions for these characteristics and validates them through finite element simulation analysis using a TTS-300-2 synchronous condenser as a case study.The simulation results align with the theoretical derivations, leading to the following conclusions: 1.The occurrence of an inter-turn short circuit in the excitation winding distorts the air gap magnetic flux density.This distortion is further amplified by unbalanced grid voltage, resulting in increased losses in the air gap magnetic flux density.When both unbalanced grid voltage and inter-turn short in the excitation winding are present, the loss of air gap magnetic flux density is even more significant compared to the case of a single fault.2. In the absence of an inter-turn short circuit in the excitation winding, the impact of unbalanced grid voltage on the circulating current between stator parallel branches can be disregarded.However, when compound faults occur, both the degree of unbalanced grid voltage and the severity of the short circuit can result in fluctuations in the circulating current between stator parallel branches, with the severity of the short circuit having a more significant influence on the circulating current.3. When diagnosing minor faults in a synchronous condenser, relying solely on features such as the even harmonic component of the circulating current may result in inaccurate fault type determination.To enhance fault diagnosis in future research, it is recommended to incorporate electromechanical information fusion, combining mechanical and electrical characteristics, for more reliable results.
By studying the inter-turn short circuit fault in synchronous condensers under unbalanced voltage conditions, this research provides references and guidance for the safe operation of synchronous condensers.It also establishes a foundation for further research on fault diagnosis in synchronous condensers.

Conclusions
This paper examines the impact of inter-turn short circuits in the excitation winding and unbalanced grid voltage on the air gap magnetic flux density and circulating current between stator parallel branches in a synchronous condenser.The study presents derived mathematical expressions for these characteristics and validates them through finite element simulation analysis using a TTS-300-2 synchronous condenser as a case study.The simulation results align with the theoretical derivations, leading to the following conclusions: 1.
The occurrence of an inter-turn short circuit in the excitation winding distorts the air gap magnetic flux density.This distortion is further amplified by unbalanced grid voltage, resulting in increased losses in the air gap magnetic flux density.When both unbalanced grid voltage and inter-turn short circuit in the excitation winding are present, the loss of air gap magnetic flux density is even more significant compared to the case of a single fault.2.
In the absence of an inter-turn short circuit in the excitation winding, the impact of unbalanced grid voltage on the circulating current between stator parallel branches can be disregarded.However, when compound faults occur, both the degree of unbalanced grid voltage and the severity of the short circuit can result in fluctuations in the circulating current between stator parallel branches, with the severity of the short circuit having a more significant influence on the circulating current.

3.
When diagnosing minor faults in a synchronous condenser, relying solely on features such as the even harmonic component of the circulating current may result in inaccurate fault type determination.To enhance fault diagnosis in future research, it is recommended to incorporate electromechanical information fusion, combining mechanical and electrical characteristics, for more reliable results.
By studying the inter-turn short circuit fault in synchronous condensers under unbalanced voltage conditions, this research provides references and guidance for the safe operation of synchronous condensers.It also establishes a foundation for further research on fault diagnosis in synchronous condensers.

Figure 1 .
Figure 1.A phase winding equivalent circuit diagram.

Figure 1 .
Figure 1.A phase winding equivalent circuit diagram.

Figure 2 .Figure 3 .
Figure 2. Two-dimensional finite element model of new type synchronous condenser.

Figure 6 .
Figure 6.A-phase winding terminal voltage and stator current.

Figure 6 .
Figure 6.A-phase winding terminal voltage and stator current.

Figure 6 .
Figure 6.A-phase winding terminal voltage and stator current.

Figure 6 .
Figure 6.A-phase winding terminal voltage and stator current.

Figure 7 .
Figure 7. (a) Air gap flux density of different short circuit degrees under grid voltage balance; (b) Harmonic analysis of air gap flux density with different short circuit degrees under grid voltage balance.

Figure 7 .Figure 8 .
Figure 7. (a) Air gap flux density of different short circuit degrees under grid voltage balance; (b) Harmonic analysis of air gap flux density with different short circuit degrees under grid voltage balance.

Figure 8 .
Figure 8.(a) Stator parallel circulation with different short circuit degrees when grid voltage is balanced; (b) Harmonic analysis of stator parallel circulating current with different short circuit degrees under grid voltage balance.

Figure 9 .
Figure 9. (a) Fault-free air gap flux density of synchronous condenser under unbalanced grid voltage; (b) Harmonic analysis of fault-free air gap flux density of synchronous condenser under unbalanced grid voltage.

Figure 9 .
Figure 9. (a) Fault-free air gap flux density of synchronous condenser under unbalanced grid voltage; (b) Harmonic analysis of fault-free air gap flux density of synchronous condenser under unbalanced grid voltage.

Figure 10 .
Figure 10.Parallel circulating current of fault-free stator of synchronous condenser under unbalanced grid voltage.3.2.3.Analysis of Air Gap Magnetic Flux Density and Stator Parallel Circulating Current under Unbalanced Grid Voltage with Excitation Winding Turn-to-Turn Short Circuit•Analysis of Air Gap Magnetic Flux Density Figure11a,b visually represent the air gap magnetic flux density under different degrees of turn-to-turn short circuits in the excitation winding when the grid voltage is unbalanced.Similarly, Figure12a,b depict the analysis results of air gap magnetic flux density under varying levels of grid voltage imbalance with excitation winding turn-to-turn short circuits.The observations from these figures reveal that unbalanced grid voltage introduces distortions in the air gap magnetic flux density, which are further amplified in the presence of turn-to-turn short circuit faults in the excitation winding.Consequently, there is a significant loss of air gap magnetic flux density.Furthermore, the combined occurrence of faults exacerbates the loss of air gap magnetic flux density compared to a single fault scenario.As the degree of turn-to-turn short circuit in the excitation winding intensifies, even harmonics, particularly the second harmonic, become more prominent in the air gap magnetic flux density.Conversely, grid voltage imbalance primarily affects the fundamental and third harmonic components of the air gap magnetic flux density.The fundamental magnetic flux density decreases while the third harmonic magnetic flux density increases with increasing voltage imbalance.These factors collectively influence the various harmonic components.The simulation results align with the theoretical analysis conducted.

Figure 11 .
Figure 11.(a) A total of 90% rated voltage different short circuit turns air gap flux density; (b) Air gap flux density harmonic analysis of 90% rated voltage with different short circuit turns.

Figure 10 .
Figure 10.Parallel circulating current of fault-free stator of synchronous condenser under unbalanced grid voltage.3.2.3.Analysis of Air Gap Magnetic Flux Density and Stator Parallel Circulating Current under Unbalanced Grid Voltage with Excitation Winding Turn-to-Turn Short Circuit•Analysis of Air Gap Magnetic Flux Density Figure11a,b visually represent the air gap magnetic flux density under different degrees of turn-to-turn short circuits in the excitation winding when the grid voltage is unbalanced.Similarly, Figure12a,b depict the analysis results of air gap magnetic flux density under varying levels of grid voltage imbalance with excitation winding turn-to-turn short circuits.The observations from these figures reveal that unbalanced grid voltage introduces distortions in the air gap magnetic flux density, which are further amplified in the presence of turn-to-turn short circuit faults in the excitation winding.Consequently, there is a significant loss of air gap magnetic flux density.Furthermore, the combined occurrence of faults exacerbates the loss of air gap magnetic flux density compared to a single fault scenario.As the degree of turn-to-turn short circuit in the excitation winding intensifies, even harmonics, particularly the second harmonic, become more prominent in the air gap magnetic flux density.Conversely, grid voltage imbalance primarily affects the fundamental and third harmonic components of the air gap magnetic flux density.The fundamental magnetic flux density decreases while the third harmonic magnetic flux density increases with increasing voltage imbalance.These factors collectively influence the various harmonic components.The simulation results align with the theoretical analysis conducted.

Energies 2023 , 20 Figure 10 .
Figure 10.Parallel circulating current of fault-free stator of synchronous condenser under unbalanced grid voltage.3.2.3.Analysis of Air Gap Magnetic Flux Density and Stator Parallel Circulating Current under Unbalanced Grid Voltage with Excitation Winding Turn-to-Turn Short Circuit•Analysis of Air Gap Magnetic Flux Density Figure11a,b visually represent the air gap magnetic flux density under different degrees of turn-to-turn short circuits in the excitation winding when the grid voltage is unbalanced.Similarly, Figure12a,b depict the analysis results of air gap magnetic flux density under varying levels of grid voltage imbalance with excitation winding turn-to-turn short circuits.The observations from these figures reveal that unbalanced grid voltage introduces distortions in the air gap magnetic flux density, which are further amplified in the presence of turn-to-turn short circuit faults in the excitation winding.Consequently, there is a significant loss of air gap magnetic flux density.Furthermore, the combined occurrence of faults exacerbates the loss of air gap magnetic flux density compared to a single fault scenario.As the degree of turn-to-turn short circuit in the excitation winding intensifies, even harmonics, particularly the second harmonic, become more prominent in the air gap magnetic flux density.Conversely, grid voltage imbalance primarily affects the fundamental and third harmonic components of the air gap magnetic flux density.The fundamental magnetic flux density decreases while the third harmonic magnetic flux density increases with increasing voltage imbalance.These factors collectively influence the various harmonic components.The simulation results align with the theoretical analysis conducted.

Figure 11 .
Figure 11.(a) A total of 90% rated voltage different short circuit turns air gap flux density; (b) Air gap flux density harmonic analysis of 90% rated voltage with different short circuit turns.Figure 11.(a) A total of 90% rated voltage different short circuit turns air gap flux density; (b) Air gap flux density harmonic analysis of 90% rated voltage with different short circuit turns.

Figure 11 .
Figure 11.(a) A total of 90% rated voltage different short circuit turns air gap flux density; (b) Air gap flux density harmonic analysis of 90% rated voltage with different short circuit turns.Figure 11.(a) A total of 90% rated voltage different short circuit turns air gap flux density; (b) Air gap flux density harmonic analysis of 90% rated voltage with different short circuit turns.

Figure 12 .
Figure 12.(a) Air gap flux density of 5-turn short circuit with different voltage balance; (b) Harmonic analysis of air gap flux density of 5-turn short circuit with different voltage balance.

Figure 12 .
Figure 12.(a) Air gap flux density of 5-turn short circuit with different voltage balance; (b) Harmonic analysis of air gap flux density of 5-turn short circuit with different voltage balance.

Figure 12 .
Figure 12.(a) Air gap flux density of 5-turn short circuit with different voltage balance; (b) Harmonic analysis of air gap flux density of 5-turn short circuit with different voltage balance.

Figure 13 .
Figure 13.(a) A total of 90% rated voltage stator parallel circulation with different short circuit turns; (b) 90% rated voltage different short circuit turns stator parallel circulation harmonic analysis.

Figure 13 .Figure 14 .
Figure 13.(a) A total of 90% rated voltage stator parallel circulation with different short circuit turns; (b) 90% rated voltage different short circuit turns stator parallel circulation harmonic analysis.

Figure 14 .
Figure 14.(a) Five-turn short circuit stator parallel circulation with different voltage balance; (b) Harmonic analysis of five-turn short circuit stator parallel circulation with different voltage balance.

Author Contributions:
Conceptualization and methodology, J.L. and C.Z.; software, C.Z. and X.H.; writing-original draft preparation, J.L. and C.Z.; writing-review and editing, J.L., C.Z., J.G. and Y.M.; visualization, J.L. and Y.H.; supervision, resources, project administration, J.L. and Y.H.; funding acquisition, J.L. and Y.H.All authors have read and agreed to the published version of the manuscript.

Table 1 .
The main information of Mmesh division in 2D finite element model.Figures 2 and 3 illustrate the model and circuit schematic representations, respectively.The critical parameters of the synchronous condenser are provided in Table

Table 2 .
Parameters of New Synchronous Condenser.

Table 3 .
Comparison of actual data and simulation data.

Table 3 .
Comparison of actual data and simulation data.

Table 3 .
Comparison of actual data and simulation data.

Table 3 .
Comparison of actual data and simulation data.

Table 4 .
The values of air gap magnetic flux density at different time instants under 90% rated voltage.

Table 4 .
The values of air gap magnetic flux density at different time instants under 90% rated voltage.