Table 1 Comparison between the developed method and multiple existing methods.

1- Required measurements

• Two voltage signals are converted into discrete values to estimate the coherence coefficients, which are measured using two voltage transformers installed at two different points of the same AC machine stator winding. Each stator winding has a center point and a full point for the voltage measurements. As a consequence, data processing and transmission are executed swiftly

• To protect AC machines, most existing protection methods required the measurement of three-phase currents^{12,19,34,35,36,37}, others needed three-phase voltages and currents^39,40, and fewer methods necessitated three-phase voltages^33,45. But certain methods do not require any voltage or current signature^10,11,32. For three-phase machines, the majority of measurements are acquired at neutral, load/supply, or both terminals

2- Main concept

• The approach relies on the cross-coherence and auto-coherence coefficients, which can be quantified for the two voltage signals for each phase

• Most methods detected the turn-to-turn and shunt faults by estimating the RMS values of current^{33,34,35,36,37}, voltage³³, power²⁴, running frequency²⁰, impedance, or a combination of these variables⁴⁶, which affect the main parameters of a machine’s power quality

3- Protection tripping characteristic curves

• The external and internal faults can be distinguished using the restraining and tripping regions, respectively, included in the designed quadratic characteristic curves based on the cross-coherence and auto-coherence coefficients, where their values lie between zero and one per unit

• Most relay characteristic curves have an open form, their boundaries are not bounded, and their settings are contingent on the parameters of the AC machines ^{33,34,35,36,37,38}. Moreover, instrument transformer specifications have an impact on the relay setting/characteristics^{33,34,35,36,37,38}

4- Protection settings

• The numerical values of the data window and the coherence settings can be used to modify the attributes of the relay (such as sensitivity, dependability, security, and speed)

• No calculation of relay settings is required. Tripping and restraining regions within the characteristic curves can be altered using the coherence settings, and varying a machine size does not affect these settings

• Multiple existing protection techniques need in-depth studies to equip their settings accurately^{33,34,35,36,37,38}

• A variation in the machine size leads to a necessary modification in the relay settings^{33,34,35,36,37,38}

5- Multiple functions

• The coherence coefficients for voltage signals can be used to implement several functions as follows:

(1) The identification of shunt faults,

(2) The determination of inter-turn faults, and the estimation of their convenient operating times when they are present,

(3) The classification of ten shunt fault types,

(4) The selection of the phases that are faulty,

(5) The discrimination between shunt faults located within and out of the equipment protection zone, and

(6) The detection and assessment of the three-phase voltages that are unbalanced,

• Many methods only run one or two protection functions. In^{10,11,32,33,36,39,40}, the approaches are unable to perform the following functions:

(1) The categorization of ten shunt fault types,

(2) he discrimination of the phases that are faulty, and

(3) The distinction between shunt faults that happen inside or outside the machine protection zone

6- Response to inter-turn faults

• The proposed scheme responds to the occurrence of inter-turn faults

• Several strategies become inactive when inter-turn faults occur^34,35,36

7- Harmonics filtering

• The utilization of the data window concept serves as a digital low-pass filter, removing some ripples and harmonics from voltage measurements

• Some approaches need additional low pass filters to remove undesirable harmonics from input signals to the relay, and relieve the effect of very fast transients in the measurements³³

8- Protection Redundancy

• The computational technique based on the coherence function reinforces the redundancy and reliability properties of the protection system. This is because it employs both auto-coherence and cross-coherence algorithms

• Several existing techniques lack the protection redundancy and reliability features^10,11,32,36

9- Protection integration

• The numerical method can be incorporated with other existing protection techniques to boost the protection reliability

• The protection integration characteristic is not involved in multiple techniques, resulting in a lack of the protection reliability^10,11,32

10- Protection speed

• The developed algorithm is highly responsive

• Some strategies have a remarkable delay in identifying turn-to-turn and shunt faults^39,40

11- Protection accuracy

• The advanced method exhibits precision in fault detection. The quantitative results illustrate that it can detect most fault situations accurately. Furthermore, it is capable of estimating the suitable tripping time in the case of turn-to-turn faults, depending on its severity

• Several existing methods have a shortage of the accuracy for inter-turn faults detection^10,11,32

12- Protection security

• The protection security can be enhanced by the excess of the data window or the coherence settings. This extends the blocking area and reduces the tripping area within the developed characteristic curves based on the coherence coefficients, simultaneously

• The security of differential or restricted earth fault current relays can be raised by maximizing the threshold values of the differential current, as well as the restraining factor^10,11. This will decrease the size of the tripping area and increase the size of the restraining area within their characteristic curves

13- Protection dependability

• The protection dependability can be improved by the reduction in the data window or the coherence settings. This lessens the blocking area and expands the tripping area within the developed characteristic curves at the same time

• The dependability of differential or restricted earth fault current relays can be reinforced by reducing the threshold values of the differential current, as well as the restraining factor^10,11,36. This will extend the size of the tripping area and reduce the size of the restraining area within their characteristic curves

14- Protection sensitivity

• The reduction of the data window or the coherence setting deviations of the proposed characteristic curves can enhance the protection sensitivity. The setting deviations can be used to alter the blocking and tripping areas situated within the developed characteristic curves, thereby modifying the sensitivity of the protection

• Each characteristic curve based on the coherence coefficients has a stationary size and extends from 0.0 to + 1.0 per unit. Thus, if the blocking area is decreased, the tripping area will be increased within each characteristic curve

• The sensitivity of differential or restricted earth fault current relays can be improved by minimizing the threshold values of the differential current, as well as the restraining factor^34,35,36,37. This will increase the tripping area size and decrease the blocking area size within their characteristic curves

• Only a single curve can be selected for these relays. Accordingly, the protection sensitivity of these relays cannot be automatically adjusted^34,35,36,37

• The settings calculation of the operating characteristic curves for these relays relies on the parameters of AC machines and the specifications of current transducers^34,35,36,37

15- Various Applications

• The protection algorithm based on the coherence estimators can be applied for a diversity of AC machine types and sizes. It can be used to find diverse types of faults that occur on the windings of the machines with different power and voltage scales. Furthermore, it can be used to create a protection strategy to protect power transformer windings. The method can be effective for protecting single-phase and three-phase machines as well. It can also be applied to protect the AC machines with one or two windings per phase

• Several existing techniques can be used to protect different components of traditional and smart power systems with a wide range of power and voltage ratings. They can be employed to protect AC generators^{5,9,31,33,38,39,40,44}, motors^{1,2,3,4,6,7,8,10,11,12,13,14,15,18,19,20,21,22,23,24,25,30,32,45} and power transformers^{26,27,28,29,34,35,36,37} from shunt or inter-turn faults

16- Financial requirements

• The proposal has lower financial commitments than some published methods

• Certain existing approaches necessitate more financial requirements^10,11,32