A Study on the Out-of-Step Detection Algorithm Using Time Variation of Complex Power-Part II: Out-of-Step Detection Algorithm and Simulation Results

: One of the established unstable power swing (out-of-step) detection algorithms in micro grid / smart grid power systems uses a trajectory of apparent impedance in the R-X plane. However, this algorithm is not suitable for fast out-of-step conditions and it is hard to detect out-of-step conditions exactly. Another algorithm for out-of-step detection is using phasor measurement units (PMUs). However, PMUs need extra equipment. This paper presents the out-of-step detection algorithm using the trajectory of complex power. The trajectory of complex power and generator mechanical power is used to identify out-of-step conditions. A second order low pass digital ﬁlter is used to extract the generator mechanical power from the complex power. Variations of complex power are used to identify equilibrium points between stable and unstable conditions. The proposed out-of-step algorithm is based on the modiﬁcation of assessment of a transient stability using equal area criterion (EAC). The proposed out-of-step algorithm is veriﬁed and tested by using alternative transient program / electromagnetic transient program (ATP / EMTP) MODELS.


Introduction
The trajectory of apparent impedance has been used to detect out-of-step conditions like micro grid/smart grids in power systems. However, a fast out-of-step condition may pass the out-of-step detection zone very fast, so it is hard to detect fast out-of-step conditions and one cannot detect out-of-step conditions exactly. Another out-of-step detection algorithm is using a phasor measurement unit (PMU) [1][2][3][4]. By measuring a phasor designed at a relay, a PMU can estimate the generator angle directly. However, to use PMU a Global Positioning System (GPS) is necessary. From all the above cases, it becomes apparent that a new out-of-step detection algorithm is needed which should consider transient stability in power systems.
This paper describes the new out-of-step detection algorithm which uses the trajectory of complex power and generator mechanical power. The generator mechanical power is extracted from the reactive power. Transient stability is estimated by complex power and mechanical power with complex power mathematical model which was mentioned in Part I. The new out-of-step detection algorithm which is described this paper could replace conventional out-of-step detection algorithms and it is faster and more accurate. The proposed algorithm is verified and tested by ATP/EMTP MODELS. Figure 1 shows out-of-step detection algorithm using a single blinder in the R-X plane. Each blinder is located on both sides of 100% of the line impedance. When an out-of-step condition occurs, the convergence speed of the apparent impedance is slower than that of a fault condition. Equation (1) shows the condition for detecting an unstable swing [5]: |T2 − T1| > T tripping (1) where T1 and T2 are the blinder passing time of the apparent impedance on the R-X plane and T tripping is the threshold time of tripping. If the passing time between T1 and T2 is larger than T tripping , it is an unstable swing. However, this algorithm cannot distinguish between faults and stable power swings and cannot detect unstable swings until the trajectory of the apparent impedance passes T2.
The generator angle can be increased [5].
Energies 2020, 13, x FOR PEER REVIEW 2 of 15 The distance relay uses apparent impedance to detect power swing conditions in the R-X plane, but when a power swing occurs, the apparent impedance trajectory is similar to a fault condition, so every case which uses apparent impedance to identify power swing conditions has to focus on distinguishing between faults and power swings. When a fault occurs in a power system, the apparent impedance converges on the fault detection zone fast. Likewise, if a power swing occurs in a power system, the apparent impedance converges on the fault detection zone.
However, when a power swing occurs, the convergence speed of the apparent impedance is slower than under fault conditions, so the convergence speed can be a reference to discriminate between faults and power swing [5][6][7]. Power swings and faults in power systems may disturb the power supply and cause damage in micro grid/smart grid power systems. From such a view point, these systems must be protected from power swings and faults. To do so power swings should generally, be blocked and faults should be tripped. Figure 1 shows out-of-step detection algorithm using a single blinder in the R-X plane. Each blinder is located on both sides of 100% of the line impedance. When an out-of-step condition occurs, the convergence speed of the apparent impedance is slower than that of a fault condition. Equation (1) shows the condition for detecting an unstable swing [5]:

Out-of-Step Detection Algorithm Using a Single Blinder in R-X Plane
where T1 and T2 are the blinder passing time of the apparent impedance on the R-X plane and Ttripping is the threshold time of tripping. If the passing time between T1 and T2 is larger than Ttripping, it is an unstable swing. However, this algorithm cannot distinguish between faults and stable power swings and cannot detect unstable swings until the trajectory of the apparent impedance passes T2. The generator angle can be increased [5].

Out-of-Step Detection Algorithm Using a Double Blinder in the R-X Plane
Likewise, Figures 1 and 2 show out-of-step detection algorithms using the apparent impedance in the R-X plane. Different from the previous algorithm, it uses a double blinder instead of a single blinder and it can identify stable swings. The power swing detection element "dZ" in Figure 2 is used as a reference of identification which distinguishes between stable swings and unstable swings.

Out-of-Step Detection Algorithm Using a Double Blinder in the R-X Plane
Likewise, Figures 1 and 2 show out-of-step detection algorithms using the apparent impedance in the R-X plane. Different from the previous algorithm, it uses a double blinder instead of a single blinder and it can identify stable swings. The power swing detection element "dZ" in Figure 2 is used as a reference of identification which distinguishes between stable swings and unstable swings.  Figure 2. Out-of-step detection algorithm using a double blinder in the R-X plane.
Referring to Equation (2), faults and power swings can be identified by: where TdZ: Passing time of apparent impedance and TPSB: Threshold time of power swing blocking. Power swing and faults can be identified by the speed of the apparent impedance which passes through the power swing detection element. Stable swings and unstable swings can also be identified by dZ. If an unstable swing occurs, the trajectory of the apparent impedance would pass from right dZ to left dZ (left dZ is the left side of Zone 1 and right dZ is the right side of Zone 1) [5][6][7]. However, if a stable swing occurs in the power system, the trajectory of apparent power will curve instead of passing left dZ. This algorithm also has to wait until the trajectory of the apparent impedance passes the left side dZ to identify an unstable swing.

Out-of-Step Detection Algorithm Using a Concentric Circle Type in the R-X Plane
The concentric circle type which uses Mho characteristics in a distance relay could be used to detect out-of-step conditions. Figure 3 shows how to detect unstable and stable swings in the R-X plane. It has the ability to detect the trajectory of apparent impedance from above or below. It also uses the apparent impedance, but this algorithm also has to wait until the trajectory of the apparent impedance has passed the left side dZ to identify unstable swings [5][6][7].   Referring to Equation (2), faults and power swings can be identified by: where T dZ : Passing time of apparent impedance and T PSB : Threshold time of power swing blocking. Power swing and faults can be identified by the speed of the apparent impedance which passes through the power swing detection element. Stable swings and unstable swings can also be identified by dZ. If an unstable swing occurs, the trajectory of the apparent impedance would pass from right dZ to left dZ (left dZ is the left side of Zone 1 and right dZ is the right side of Zone 1) [5][6][7]. However, if a stable swing occurs in the power system, the trajectory of apparent power will curve instead of passing left dZ. This algorithm also has to wait until the trajectory of the apparent impedance passes the left side dZ to identify an unstable swing.

Out-of-Step Detection Algorithm Using a Concentric Circle Type in the R-X Plane
The concentric circle type which uses Mho characteristics in a distance relay could be used to detect out-of-step conditions. Figure 3 shows how to detect unstable and stable swings in the R-X plane. It has the ability to detect the trajectory of apparent impedance from above or below. It also uses the apparent impedance, but this algorithm also has to wait until the trajectory of the apparent impedance has passed the left side dZ to identify unstable swings [5][6][7].  Figure 2. Out-of-step detection algorithm using a double blinder in the R-X plane.
Referring to Equation (2), faults and power swings can be identified by: where TdZ: Passing time of apparent impedance and TPSB: Threshold time of power swing blocking. Power swing and faults can be identified by the speed of the apparent impedance which passes through the power swing detection element. Stable swings and unstable swings can also be identified by dZ. If an unstable swing occurs, the trajectory of the apparent impedance would pass from right dZ to left dZ (left dZ is the left side of Zone 1 and right dZ is the right side of Zone 1) [5][6][7]. However, if a stable swing occurs in the power system, the trajectory of apparent power will curve instead of passing left dZ. This algorithm also has to wait until the trajectory of the apparent impedance passes the left side dZ to identify an unstable swing.

Out-of-Step Detection Algorithm Using a Concentric Circle Type in the R-X Plane
The concentric circle type which uses Mho characteristics in a distance relay could be used to detect out-of-step conditions. Figure 3 shows how to detect unstable and stable swings in the R-X plane. It has the ability to detect the trajectory of apparent impedance from above or below. It also uses the apparent impedance, but this algorithm also has to wait until the trajectory of the apparent impedance has passed the left side dZ to identify unstable swings [5][6][7].  Figure 3. Out-of-step detection algorithm using a concentric circle type in the R-X Plane. Figure 3. Out-of-step detection algorithm using a concentric circle type in the R-X Plane.

Out-of-Step Detection Algorithm Using Phasor Measurement Unit (PMU)
Unlike the previous methods, a PMU takes data from a local relay to extract the phase angle directly. Figure 4 shows data synchronization using a Global Positioning System (GPS) between two local relays. This synchronized data is then sent to a central control center (CCC) to estimate the generator angle. By using this generator angle which is estimated from data, local relays can trip faults that cause unstable swings. Because the PMU uses the generator angle instead of the trajectory of apparent impedance, it is more exact than the previous methods described in Sections 2.1 and 2.2 However, a PMU needs extra equipment. In addition, from a utility's point of view, it is best to perform the same role using an existing device without installing additional equipment. Installation of new devices can increase the complexity of the system, which is not preferred by operators. For example, it is possible to quickly share information using communication technology, but there is also another problem of installing communication systems for sharing information and paying attention to security. At this point, the new out-of-step detection algorithm that is more exact than upper the algorithm, faster than Sections 2.1 and 2.2 in the R-X plane and is low-cost (no extra equipment is needed) is required.

Out-of-Step Detection Algorithm Using Phasor Measurement Unit (PMU)
Unlike the previous methods, a PMU takes data from a local relay to extract the phase angle directly. Figure 4 shows data synchronization using a Global Positioning System (GPS) between two local relays. This synchronized data is then sent to a central control center (CCC) to estimate the generator angle. By using this generator angle which is estimated from data, local relays can trip faults that cause unstable swings. Because the PMU uses the generator angle instead of the trajectory of apparent impedance, it is more exact than the previous methods described in Sections 2.1 and 2.2 However, a PMU needs extra equipment. In addition, from a utility's point of view, it is best to perform the same role using an existing device without installing additional equipment. Installation of new devices can increase the complexity of the system, which is not preferred by operators. For example, it is possible to quickly share information using communication technology, but there is also another problem of installing communication systems for sharing information and paying attention to security. At this point, the new out-of-step detection algorithm that is more exact than upper the algorithm, faster than Sections 2.1 and 2.2 in the R-X plane and is low-cost (no extra equipment is needed) is required.   Figure 5 shows the trajectory of complex power on the complex power plane which is presented in Part Ι. As mentioned before in Part Ι, the trajectory of complex power will be moved in line with an ellipse. Mechanical power is the input power of a generator [8][9][10][11]. The point which is (0, α) is the horizontal axis of ellipse will have a 90° generator angle. This can be defined as follows:

Transient Stability Using Complex Power Variations
The reactive power Equation (3) could be written more simply as follows: If the generator angle δ is 90 degrees, then Equation (4) is written as:  Figure 5 shows the trajectory of complex power on the complex power plane which is presented in Part I. As mentioned before in Part I, the trajectory of complex power will be moved in line with an ellipse. Mechanical power is the input power of a generator [8][9][10][11]. The point which is (0, α) is the horizontal axis of ellipse will have a 90 • generator angle. This can be defined as follows:

Transient Stability Using Complex Power Variations
The reactive power Equation (3) could be written more simply as follows: If the generator angle δ is 90 degrees, then Equation (4) is written as: If the power system is stable, the mechanical power and electrical power will be same at the stable equilibrium point (SEP) [8][9][10][11]. When faults occur in a power system, the trajectory of complex power will be swinging and may pass an unstable equilibrium point (UEP), i.e., the point at which the equilibrium collapses as the generator's angle increases by 90 • , then the power system will be in an unstable swing condition (out-of-step condition).
Energies 2020, 13, x FOR PEER REVIEW 5 of 15 If the power system is stable, the mechanical power and electrical power will be same at the stable equilibrium point (SEP) [8][9][10][11]. When faults occur in a power system, the trajectory of complex power will be swinging and may pass an unstable equilibrium point (UEP), i.e., the point at which the equilibrium collapses as the generator's angle increases by 90°, then the power system will be in an unstable swing condition (out-of-step condition).

Mechnical Power Estimation from Electrical Power
To determine SEP or UEP, a reference is needed. At this point, mechanical power can be used to determine stable or unstable conditions. If the mechanical power and electrical power are the same with a generator angle over 90°, it will be an unstable condition [8][9][10][11]. Otherwise, an generator angle below 90°, indicates a stable condition.
However, it is not easy to take generator angle data directly from generators, so another way to take generator angles is needed. As mentioned before, mechanical power and electrical power are the same at a SEP. This means that the mechanical power can be estimated from the electrical power at the SEP. There are two ways to estimate mechanical power from electrical power. One is using electrical power data before a power swing. Another is mechanical power estimation from electrical power using a low-pass filter. This paper uses mechanical power data estimation from electrical power using a second order Butterworth low-pass filter, as shown in Figure 6.

Mechnical Power Estimation from Electrical Power
To determine SEP or UEP, a reference is needed. At this point, mechanical power can be used to determine stable or unstable conditions. If the mechanical power and electrical power are the same with a generator angle over 90 • , it will be an unstable condition [8][9][10][11]. Otherwise, an generator angle below 90 • , indicates a stable condition.
However, it is not easy to take generator angle data directly from generators, so another way to take generator angles is needed. As mentioned before, mechanical power and electrical power are the same at a SEP. This means that the mechanical power can be estimated from the electrical power at the SEP. There are two ways to estimate mechanical power from electrical power. One is using electrical power data before a power swing. Another is mechanical power estimation from electrical power using a low-pass filter. This paper uses mechanical power data estimation from electrical power using a second order Butterworth low-pass filter, as shown in Figure 6. take generator angles is needed. As mentioned before, mechanical power and electrical power are the same at a SEP. This means that the mechanical power can be estimated from the electrical power at the SEP. There are two ways to estimate mechanical power from electrical power. One is using electrical power data before a power swing. Another is mechanical power estimation from electrical power using a low-pass filter. This paper uses mechanical power data estimation from electrical power using a second order Butterworth low-pass filter, as shown in Figure 6.   Figure 6 shows how to estimate the mechanical power from the electrical power using a second order low-pass filter. To estimate mechanical power from electrical power, the cut-off frequency of the low-pass filter must be determined. This cut-off frequency can be determined using frequency spectrum analysis in the frequency domain. Figure 7 shows a spectrum analysis of electrical power in the frequency domain to determine the low-pass filter cut-off frequency. Before the power swing condition, electrical power which contains only a DC component in the frequency domain operates in a steady state. Therefore, the cut-off frequency has to determine how to extract this DC component in the frequency domain. In order to identify the cut-off frequency of the low-pass filter, simulation studies have been conducted on various events, such as line length variations and different fault locations.
Energies 2020, 13, x FOR PEER REVIEW 6 of 15 Figure 6 shows how to estimate the mechanical power from the electrical power using a second order low-pass filter. To estimate mechanical power from electrical power, the cut-off frequency of the low-pass filter must be determined. This cut-off frequency can be determined using frequency spectrum analysis in the frequency domain Figure 7 shows a spectrum analysis of electrical power in the frequency domain to determine the low-pass filter cut-off frequency. Before the power swing condition, electrical power which contains only a DC component in the frequency domain operates in a steady state. Therefore, the cutoff frequency has to determine how to extract this DC component in the frequency domain. In order to identify the cut-off frequency of the low-pass filter, simulation studies have been conducted on various events, such as line length variations and different fault locations.  Table 1 shows that the simulation result of the cut-off frequency is 0.1 Hz and this is used to estimate the mechanical power from electrical power in this paper. Based on various simulations that were performed, the cut-off frequency was 0.1 Hz when considering the implementation of the filter for relay and ATP/EMTP simulation.  Figure 8 shows the out-of-step detection algorithm using a time variation of complex power. There are two points Qmax and Q0. Qmax is the reactive power of UEP and Q0 is the reactive power of SEP. Based on the Part I, the equilibrium points can classify Qmax and Q0 by α, which is the horizontal axis of the complex power ellipse. However, as shown in Equation (5), if α is used to classify equilibrium points, line data and source data (each side) are needed to solve Equation (5), thus another method is needed,  Table 1 shows that the simulation result of the cut-off frequency is 0.1 Hz and this is used to estimate the mechanical power from electrical power in this paper. Based on various simulations that were performed, the cut-off frequency was 0.1 Hz when considering the implementation of the filter for relay and ATP/EMTP simulation.  Figure 8 shows the out-of-step detection algorithm using a time variation of complex power. There are two points Q max and Q 0 . Q max is the reactive power of UEP and Q 0 is the reactive power of SEP. Based on the Part I, the equilibrium points can classify Q max and Q 0 by α, which is the horizontal axis of the complex power ellipse. However, as shown in Equation (5), if α is used to classify equilibrium points, line data and source data (each side) are needed to solve Equation (5), thus another method is needed, so instead of reference α, the time variation of complex power with the equilibrium point is used to identify out-of-step condition. Figure 8 shows how to classify four cases which are shown in Figure 8.  Δ are used to classify each case. By using this, the state of the power system (unstable swing or stable swing) can be identified and out-of-step conditions can be detected. As shown above, ① and ② are stable swings and ③ and ④ are unstable swing conditions. The final goal of this paper is a new out-of-step detection algorithm, so stable swing will not be    Δ are used to classify each case. By using this, the state of the power system (unstable swing or stable swing) can be identified and out-of-step conditions can be detected. As shown above, ① and ② are stable swings and ③ and ④ are unstable swing conditions. The final goal of this paper is a new out-of-step detection algorithm, so stable swing will not be discussed. Case ④ is an unstable swing after out-of-step condition, so it will also not be discussed. Figure 10 shows the novel out-of-step detection algorithm using the time variation of complex power, where ΔP is the time variation of active power and ΔQ is the time variation of reactive power. The trip signal is activated when the mechanical power Pm is in the middle of the electrical power Pn and Pn-2 which are two samples before Pn with a time variation ΔP smaller than 0 and ΔQ is larger than As shown above, 1 and 2 are stable swings and 3 and 4 are unstable swing conditions. The final goal of this paper is a new out-of-step detection algorithm, so stable swing will not be discussed. Case 4 is an unstable swing after out-of-step condition, so it will also not be discussed. Figure 10 shows the novel out-of-step detection algorithm using the time variation of complex power, where ∆P is the time variation of active power and ∆Q is the time variation of reactive power. The trip signal is activated when the mechanical power P m is in the middle of the electrical power P n and P n-2 which are two samples before P n with a time variation ∆P smaller than 0 and ∆Q is larger than 0. This algorithm consists of two steps. In the first, we find out the equilibrium point and second, we identify the unstable equilibrium point. Equation (6) shows how to find the equilibrium point:

Out-of-Step Detection Algorithm Using a Trajectory of Complex Power
where P n-2 is the active power value of the two samples mentioned before, P n is the current active power and P m is the mechanical power. There are only two points which satisfy this condition: SEP and UEP, thus how to find out the unstable equilibrium point is needed. Table 2 shows four cases to find out unstable equilibrium points.
Energies 2020, 13, x FOR PEER REVIEW 8 of 15 0. This algorithm consists of two steps. In the first, we find out the equilibrium point and second, we identify the unstable equilibrium point. Equation (6) shows how to find the equilibrium point: where Pn-2 is the active power value of the two samples mentioned before, Pn is the current active power and Pm is the mechanical power. There are only two points which satisfy this condition: SEP and UEP, thus how to find out the unstable equilibrium point is needed. Table 2 shows four cases to find out unstable equilibrium points.

State
Number Table 2 can be used to identify four cases which are the trajectories of complex power on a complex power plane. The concept of 'new out-of-step detection algorithm' is the classification of the equilibrium using the time variation of complex power.

Simulation Method
A set of simulation tests were carried in a test model of a power system shown in Figure 11 which is interfaced with a model of a relay implemented using the ATP/EMTP MODELS software. This   Table 2 can be used to identify four cases which are the trajectories of complex power on a complex power plane. The concept of 'new out-of-step detection algorithm' is the classification of the equilibrium using the time variation of complex power.

Simulation Method
A set of simulation tests were carried in a test model of a power system shown in Figure 11 which is interfaced with a model of a relay implemented using the ATP/EMTP MODELS software. This simulation is performed to prove the new out-of-step algorithm which uses the time variation of complex power in power systems. To show the advantage of our new out-of-step detection algorithm, an out-of-step detection simulation which uses a single binder in the R-X plane was conducted under the same conditions. Three phase faults occurred at the distance of 50% of the transmission line and the initial generator angle is 30°. The transmission system model comprises of a total line length of 100 km; the nominal power frequency is 60 Hz. A synchronous machine (SM) card and TACS are used for the governor and excitation system of "A" nuclear power plant in Korea [12][13][14][15].

Three Phase Fault: Clearing after 10 Cycles
The novel out-of-step detection adopting the algorithm of Figure 10 is simulated. The considered fault is a 3-phase fault, which occurs 1 s after the start of the simulation. In the simulation of the new out-of-step detection algorithm, Figure 12 depicts the active, reactive and mechanical power waveforms when the out-of-step condition has occurred after fault clearing. In Figure 12, the mechanical power which is extracted from the active power is shown. In the case of a transient fault, the reactive power is increased at the out-of-step point and the active power is decreased at the same point when the out-of-step event occurred.   Figure 13 shows a part of the ellipse which is mentioned before in Part I of the paper. When an out-of-step condition occurs, the trajectory of the To show the advantage of our new out-of-step detection algorithm, an out-of-step detection simulation which uses a single binder in the R-X plane was conducted under the same conditions. Three phase faults occurred at the distance of 50% of the transmission line and the initial generator angle is 30 • . The transmission system model comprises of a total line length of 100 km; the nominal power frequency is 60 Hz. A synchronous machine (SM) card and TACS are used for the governor and excitation system of "A" nuclear power plant in Korea [12][13][14][15].

Three Phase Fault: Clearing after 10 Cycles
The novel out-of-step detection adopting the algorithm of Figure 10 is simulated. The considered fault is a 3-phase fault, which occurs 1 s after the start of the simulation. In the simulation of the new out-of-step detection algorithm, Figure 12 depicts the active, reactive and mechanical power waveforms when the out-of-step condition has occurred after fault clearing. In Figure 12, the mechanical power which is extracted from the active power is shown. In the case of a transient fault, the reactive power is increased at the out-of-step point and the active power is decreased at the same point when the out-of-step event occurred. To show the advantage of our new out-of-step detection algorithm, an out-of-step detection simulation which uses a single binder in the R-X plane was conducted under the same conditions. Three phase faults occurred at the distance of 50% of the transmission line and the initial generator angle is 30°. The transmission system model comprises of a total line length of 100 km; the nominal power frequency is 60 Hz. A synchronous machine (SM) card and TACS are used for the governor and excitation system of "A" nuclear power plant in Korea [12][13][14][15].

Three Phase Fault: Clearing after 10 Cycles
The novel out-of-step detection adopting the algorithm of Figure 10 is simulated. The considered fault is a 3-phase fault, which occurs 1 s after the start of the simulation. In the simulation of the new out-of-step detection algorithm, Figure 12 depicts the active, reactive and mechanical power waveforms when the out-of-step condition has occurred after fault clearing. In Figure 12, the mechanical power which is extracted from the active power is shown. In the case of a transient fault, the reactive power is increased at the out-of-step point and the active power is decreased at the same point when the out-of-step event occurred.  Figure 13 depicts the trajectory of the complex power and mechanical power on the complex power plane after a fault occurrence. Complex power in Figure 13 shows a part of the ellipse which is mentioned before in Part I of the paper. When an out-of-step condition occurs, the trajectory of the complex power moves in a counterclockwise direction until it reaches beyond 180°, and the complex   Figure 13 shows a part of the ellipse which is mentioned before in Part I of the paper. When an out-of-step condition occurs, the trajectory of the complex power moves in a counterclockwise direction until it reaches beyond 180 • , and the complex power pass UEP where mechanical power and electrical power are equal. When the initial generator angle is 30°, the generator angle after the three-phase fault is shown in Figure 14. Here, the fault is cleared 10 cycles after the fault occurred. As seen in Figure 13, the angle oscillations are increased and dampen out finally after the out-of-step incident occurred. The new outof-step detection algorithm is used to detect the out-of-step condition and a tripping signal appears where the generator angle is 127°.  Figure 14. Generator angle when out-of-step occurred after the fault clearing (initial generator angle is 30°, fault clearing after 10 cycles).

Three Phase Fault: Clearing after 15 Cycles
When the fault clearing time increases from 10 cycles to 15 cycles, out-of-step conditions become serious. Figure 15 depicts the active, reactive and mechanical power waveforms after a fault has occurred. Different from above, the out-of-step condition occurred faster than in Figure 15. Also in the case of that out-of-step condition, the reactive power is increased at the out-of-step point and the active power is decreased at same point when the out-of-step incident occurred. When the initial generator angle is 30 • , the generator angle after the three-phase fault is shown in Figure 14. Here, the fault is cleared 10 cycles after the fault occurred. As seen in Figure 13, the angle oscillations are increased and dampen out finally after the out-of-step incident occurred. The new out-of-step detection algorithm is used to detect the out-of-step condition and a tripping signal appears where the generator angle is 127 • . When the initial generator angle is 30°, the generator angle after the three-phase fault is shown in Figure 14. Here, the fault is cleared 10 cycles after the fault occurred. As seen in Figure 13, the angle oscillations are increased and dampen out finally after the out-of-step incident occurred. The new outof-step detection algorithm is used to detect the out-of-step condition and a tripping signal appears where the generator angle is 127°.  Figure 14. Generator angle when out-of-step occurred after the fault clearing (initial generator angle is 30°, fault clearing after 10 cycles).

Three Phase Fault: Clearing after 15 Cycles
When the fault clearing time increases from 10 cycles to 15 cycles, out-of-step conditions become serious. Figure 15 depicts the active, reactive and mechanical power waveforms after a fault has occurred. Different from above, the out-of-step condition occurred faster than in Figure 15. Also in the case of that out-of-step condition, the reactive power is increased at the out-of-step point and the active power is decreased at same point when the out-of-step incident occurred.

Three Phase Fault: Clearing after 15 Cycles
When the fault clearing time increases from 10 cycles to 15 cycles, out-of-step conditions become serious. Figure 15 depicts the active, reactive and mechanical power waveforms after a fault has occurred. Different from above, the out-of-step condition occurred faster than in Figure 15. Also in the Energies 2020, 13, 1833 11 of 15 case of that out-of-step condition, the reactive power is increased at the out-of-step point and the active power is decreased at same point when the out-of-step incident occurred.  Figure 16 depicts the trajectory of the complex power and mechanical power on the complex power plane after a fault occurred. Different from Figure 13, the trajectory is increased which means the out-of-step condition became serious.   Figure  17, the angle oscillations are increased and go out finally after the out-of-step event occurred. At the moment of the out-of-step condition, the generator angle had increased to 125° and a tripping signal which uses the new out-of-step detection algorithm appeared where the generator angle is 125°.   Figure 16 depicts the trajectory of the complex power and mechanical power on the complex power plane after a fault occurred. Different from Figure 13, the trajectory is increased which means the out-of-step condition became serious.   Figure  17, the angle oscillations are increased and go out finally after the out-of-step event occurred. At the moment of the out-of-step condition, the generator angle had increased to 125° and a tripping signal which uses the new out-of-step detection algorithm appeared where the generator angle is 125°.

Three Phase Fault: Clearing after 20 Cycles
This is the first swing unstable (FSU) condition. Because the fault clearing time increases, the outof-step has occurred after the fault was cleared. As shown in Figure 18, an out-of-step condition occurred after 3 s.   Figure 16, the trajectory is increased which means the out-of-step condition became serious.

Three Phase Fault: Clearing after 20 Cycles
This is the first swing unstable (FSU) condition. Because the fault clearing time increases, the out-of-step has occurred after the fault was cleared. As shown in Figure 18, an out-of-step condition occurred after 3 s.

Three Phase Fault: Clearing after 20 Cycles
This is the first swing unstable (FSU) condition. Because the fault clearing time increases, the outof-step has occurred after the fault was cleared. As shown in Figure 18, an out-of-step condition occurred after 3 s.   Figure 16, the trajectory is increased which means the out-of-step condition became serious.   Figure 16, the trajectory is increased which means the out-of-step condition became serious. Out-of-Step moment Figure 19. The trajectory of complex power with mechanical power on the R-X plane (fault clearing after 20 cycles). Figure 20 depicts the first swing unstable (FSU) condition which means a generator angle divergence after the first swing. Here, the fault is cleared at 20 cycles after the fault had occurred. As seen in Figure 20, the generator angle had gone out after the out-of-step event occurred. At the moment of the out-of-step condition, the generator angle had increased to 113.5° and a tripping signal which uses new out-of-step detection algorithm appeared where the generator angle is 113.5°.

Comparison of the New Out-of-Step Detection Algorithm with a Single Blinder Algorithm
Under the same conditions, the new out-of-step detection algorithm which uses the trajectory of the complex power and the conventional out-of-step detection algorithm which uses a single blinder in the R-X plane are compared. Table 3 shows the tripping time (out-of-step detection time) and generator angle which is the angle of the out-of-step moment. Table 3. Comparisons the new out-of-step detection algorithm with a single blinder algorithm. Figure 20 depicts the first swing unstable (FSU) condition which means a generator angle divergence after the first swing. Here, the fault is cleared at 20 cycles after the fault had occurred. As seen in Figure 20, the generator angle had gone out after the out-of-step event occurred. At the moment of the out-of-step condition, the generator angle had increased to 113.5 • and a tripping signal which uses new out-of-step detection algorithm appeared where the generator angle is 113.5 • . Out-of-Step moment Figure 19. The trajectory of complex power with mechanical power on the R-X plane (fault clearing after 20 cycles). Figure 20 depicts the first swing unstable (FSU) condition which means a generator angle divergence after the first swing. Here, the fault is cleared at 20 cycles after the fault had occurred. As seen in Figure 20, the generator angle had gone out after the out-of-step event occurred. At the moment of the out-of-step condition, the generator angle had increased to 113.5° and a tripping signal which uses new out-of-step detection algorithm appeared where the generator angle is 113.5°.

Comparison of the New Out-of-Step Detection Algorithm with a Single Blinder Algorithm
Under the same conditions, the new out-of-step detection algorithm which uses the trajectory of the complex power and the conventional out-of-step detection algorithm which uses a single blinder in the R-X plane are compared. Table 3 shows the tripping time (out-of-step detection time) and generator angle which is the angle of the out-of-step moment. Table 3. Comparisons the new out-of-step detection algorithm with a single blinder algorithm.

Comparison of the New Out-of-Step Detection Algorithm with a Single Blinder Algorithm
Under the same conditions, the new out-of-step detection algorithm which uses the trajectory of the complex power and the conventional out-of-step detection algorithm which uses a single blinder in the R-X plane are compared. Table 3 shows the tripping time (out-of-step detection time) and generator angle which is the angle of the out-of-step moment.  Table 3 shows what is faster and what is more stable. The generator angle of our new out-of-step detection algorithm does not go over 180 • , however the single blinder does go over. The tripping time (out-of-step detection moment) of the new out-of-step detection algorithm is faster than that of the single blinder by about 0.6 s. Therefore, in view of the transient stability and speed of detection time, the new out-of-step detection algorithm offers a further advantage.

Conclusions
This paper has presented the new out-of-step detection algorithm for improving the system stability. Herein, the new out-of-step detection algorithm is used in order to detect out-of-step conditions and enhance the transient stability. It uses complex power instead of apparent impedance to detect out-of-step conditions. Mechanical power, which is used to reference the equilibrium point, is extracted from the electrical power using a 2nd order Butterworth filter. Using by mechanical power and the trajectory of complex power in the complex power plane, the new out-of-step detection algorithm can detect equilibrium points which represent stable or unstable states of a power system. At the same time, the time variation of complex power is used to identify equilibrium points which are stable or unstable.
The simulation results show that the equilibrium points are classified accurately by using the mechanical power, the trajectory of complex power with time variation of complex power, and out-of-step conditions are detected by using the proposed out-of-step detection algorithm.
The new out-of-step detection algorithm presented herein is fast, accurate and does not need extra equipment such as GPS. This is an advantage of the new out-of-step detection algorithm over others. From a utility's point of view, it is best to perform the same task on the existing system without installing additional equipment. Finally, the new out-of-step detection algorithm presented herein will also be applicable in backup protection and system protection schemes (SPSs) for protecting generators.