INTRODUCTION
Protection of the transmission line, one of the main components of the radial electrical power system, has a central role in power system protection because transmission lines are vital elements of the network, which connects the generating plants to the load centers, also because of the long distances traversed by transmission lines over countryside. Transmission lines are subject to a majority of the faults occurring on the power system. The simplest protection system used at the lowest system voltages consists of fuses, which act as relays and circuit breakers combined. The protection system used for mediumvoltage transmission lines is somewhat simpler than that used for highvoltage and extrahighvoltage transmission lines, which provide the major bulk transmission facilities (Ranjbar and Cory, 1974; Horowitz et al., 1988).
Knowledge of the currents resulting from various types of faults is essential for the effective operation of the protection system. Faults on a power system resulting high currents must be removed in the minimum of time. If short circuits are allowed to persist on a power system for an extended period, many or all of the following undesirable effects are likely to occur (Weedy and Cory, 1999):
• 
Reduced stability margins. 

• 
Damage of the equipment that is in the vicinity of the fault
due to heavy currents or low voltages produced by short circuit. 

• 
Explosions, which may occur in equipment containing insulating
oil during a short circuit and which may cause fire resulting in a serious
hazard to personnel and damage to other equipment. 

• 
Disruptions in the entire power system service area by a succession
of protective actions taken by different protection systems, an occurrence
known as cascading. 

Which one of these effects will predominate in a given case depends upon the nature and operating conditions of the power system. With the advent of microprocessors and the integrated circuits, numerical protection devices are now the norm. The analogue quantities can be sampled and converted to digital form for numerical manipulation, by which the fault clearance can fast be obtained (Peck et al., 1989).
Digital computers, initially used in electrical power systems for offline calculations soon found their way into load dispatching centers and online applications (Koch, 1989). The appearance of microprocessors enabled ever simpler tasks to be performed at distributed locations and their use for protection and control, in electrical power systems, was logically the next step. Thus, memory and computing capacity is now available, which enabled the processing of additional parameters and also new parameters that could not be processed previously. Nowadays, digital techniques are widely used in protection engineering. Digital fault location techniques for transmission systems when digital fault recorded data are available at one terminal or two terminals is presented by Girgis and Fallon (1992). Digital protection method for power transformers based on an equivalent circuit composed of inverse inductance is developed by Inagaki et al. (1988). Digital differential protection of power transformers is discussed by Hamedani et al. (2004). And a formal software requirements specification method for digital nuclear plant protection systems is developed by Junbeom et al. (2005).
In this study, the focus is on simulation and performanceanalysis of the 3phase shortcircuits at different locations on the radial electrical power system in order to find out a new criterion for its protection with the objective of decreasing the fault clearance time and thus increasing the stability margins of the system.
RADIAL ELECTRICAL POWER SYSTEM PROTECTION
In the radial electrical power system shown in Fig. 1, the
transformer, feeding bus one from a higher voltage supply point, supplies loads
at buses 15 through 4 transmission lines. Timeovercurrent relays are used
to protect the transmission lines of this system by providing primary protection
for the line as well as the remote backup for the neighboring line. Relays
at each of the 4 buses 14 are provided to protect their respective lines as
primary protection relays and to provide remote backup protection to one line
downstream from the relay location. Selectivity here is achieved by adding time
delay to the controlling relays of circuit breakers. For providing time delay
to the circuit breakers CB_{1}, CB_{2}, CB_{3} and CB_{4}
the tripping is delayed in the following manner: CB_{4}no added time
delay; CB_{3}  0.4 s added time delay; CB_{2 } 0.8 s added
time delay; CB_{1 } 1.2 s added time delay. A step time delay generally
between 0.3 and 0.4 s is necessary to account for the time of operation of circuit
breaker and its relay operation times.
 Fig. 1: 
Oneline diagram of a radial electrical power system 

The disadvantage of the time grading principle is that the faults near the source, which have a higher fault level than faults on sections further down the line are tripped after the longest delay. The destructive potential of a fault close to the source is thus several times greater.
The timeovercurrent protection using relays with an inverse characteristic approximately does not suffer from this drawback, because their time delay reduces as the fault current increases. The practice in this respect is somewhat divided, some countries preferring definite timeovercurrent relays to avoid the more complicated grading of the inverse characteristics of the various relays in a power system, while others have standardized on inverse definite minimum time relays. Today the grading of inverse definite minimum time relays is frequently performed by computer programs (Ungrad and Wiszniewski, 1995). Thus, the objective of this research is to develop a protection system that overcomes the disadvantages of the definitetimeovercurrent protection and the complexity of the inversetimeovercurrent protection.
MODELLING OF THE STUDY ELECTRICAL RADIAL POWER SYSTEM
In this study, mathematical modelling is used as a tool to find out a new criterion that will be helpful to achieve the objective of this research. The radial electrical power system, which is to be modeled for 3phase shortcircuit current calculations is shown in Fig. 1. The highvoltage bus is assumed to be an infinite bus. The singlephase equivalent circuit is shown in Fig. 2.
To decrease the number of differential equations by one 3rd, the mathematical
model is represented in a rectangular system of coordinates (x, y) (AlJufout,
2005). The differential equations in x axis are as follows:
 Fig. 2: 
Singlephase equivalent circuit of the study radial electrical
power system 

Where, 
j 
= 
1, 2,..., 4. 
k 
= 
j + 1. 
l 
= 
1, 2, ..., 5. 

and similarly in y axis:
The voltages at the highvoltage bus in x and y coordinates can be calculated
using the 3phase values as follows:
where:
The equations of the nodal voltages for each axis can be derived from Kirchhoff’s
law for the current derivatives (Alyyan et al., 2005) as: the algebraic
sum of the current derivatives entering any node is zero:
These equations for nodal voltages determination can be represented in matrix
form as follows:
where:
where, the damping coefficients are calculated as follows:
Where, 
R_{i} 
= 
The load resistance connected to bus i. 
R_{iJ}. 
= 
The transmission line resistance connected between buses i and j 
L_{i} 
= 
The load inductance connected to bus i. 
L_{iJ} 
= 
The transmission line inductance connected between buses i and j. 
R_{T}, L_{T} 
= 
The resistance and leakage inductance of the power transformer, respectively. 

SUGGESTED OVERANGLE PROTECTION METHOD
The developedabove algorithm can be used for both steadystate and transient analyses. The system of the differential equations can be solved using 4thorder RungeKutta method and the system of the algebraic equations can be solved using gauss method. The infinite bus phase voltages are given. The initial values of all the variables (currents) are equal to zero at the first step of solution.
Table 1 shows the parameters of the loads connected to buses
15 (Fig. 2). These parameters are indicated in both actual
values and per unit (p.u.), where the base voltage is:
and the base current is:
The 3phase transformer is assumed to be rated 20 MVA, 138/20 kV with leakage reactance of 10% and resistance of 2%. The infinite bus voltage is assumed to be equal to 1.125 p.u. The transmission line resistance and inductive reactance are assumed to be 0.1284 and 0.788 Ω mi^{1}, respectively.
Table 2 shows the transmission line parameters. Based on the abovedescribed algorithm, the steadystate prefault conditions have been calculated and are shown in Table 3.
The steadystate and inrush values of the shortcircuit current have been calculated and are shown in Table 4 in both actual values and per unit. When using approximate calculation methods, where the resistances and static loads are negligible, the shortcircuit currents differ approximately by 1% (AlJufout, 2006).
Figure 3 shows the waveforms of the phase currents of the
transmission line, which is connected between buses 1 and 2, when a 3phase
fault occurs on bus 2, where the prefault current value is equal to 0.629 p.u.
and the steadystate fault current is equal to 4.88 p.u. The bold curve in Fig.
3 shows the current envelope, which is calculated as follows:
where, i_{a}, i_{b}, i_{c }are the instantaneous phase currents.
Figure 4 shows the current envelope flowing through the transmission
line, which is connected between buses 1 and 2, when a 3phase fault occurs
on buses 25. The current envelope is shown during only the first 2 ms after
the fault instant. It is clear from Fig. 4 that the inclination
angle of the current envelope depends on the fault location.
Table 1: 
Loads parameters 


Table 2: 
Transmission line parameters 


Table 3: 
Prefault loads parameters 


Table 4: 
Threephase fault currents 


Figure 5 and 7 show the current envelopes
of the other transmission lines in the study power system of Fig.
1.
Table 5 shows the inclination angles of different transmission
lines currents, when 3phase shortcircuit occurs on different buses of the
power system. This inclination angle is calculated by the following formula:
It is suggested to use this inclination angle as a criterion for protection against 3phase faults on radial electrical power systems.
Protection zones (instantaneous and time delayed) and their settings are shown
in Table 6. Calculations show that all the current envelope
changes caused by normal operating conditions occur with an inclination angle,
which is <10°. The latter fact can be used in backup protection in order
to avoid under functioning. Each protection unit provides primary and multizone
backup protection.
 Fig. 3: 
Change of the transmission line (12) phase currents and the
current envelope during fault on bus 2 

 Fig. 4: 
Changes of the transmission line (12) current envelopes during
faults on buses 25 

 Fig. 5: 
Changes of the transmission line (23) current envelopes during
faults on buses 35 

The time grading diagram of the proposed instantaneous and timeoverangle protection is shown in Fig. 8.
Since, buses are assumed as zeroimpedance power system elements and to discriminate
between faults on the receiving end of the transmission line and faults on the
sending end of the downstream transmission line as well as faults on the outgoing
load feeders, the reach of each protection zone is decreased by 5%.
 Fig. 6: 
Changes of the transmission line (34) current envelopes during
faults on buses 4 and 5 

 Fig. 7: 
Change of the envelope of the transmission line (45) current
during 3phase short circuit on bus 5 

Table 5: 
Inclination angle of the transmission line current envelope 


Table 6: 
Instantaneous and timeoverangle protection settings 


Thus, the protection zones (instantaneous and time delayed) and their settings
become as shown in Table 7.
 Fig. 8: 
Time grading diagram of the instantaneous and timeoverangle
(OA) protection 

 Fig. 9: 
Time grading diagram of the 95% instantaneous and timeoverangle
protection 

 Fig. 10: 
Simplified block diagram of instantaneous and timeoverangle
protection with one timedelayed zone 

The time grading diagram of the 95% instantaneous and timeoverangle protection is shown in Fig. 9.
To avoid over functioning, particularly when a timedelayed zone operates,
a condition of that the current envelope is greater than the current envelope
of the maximum load must be obtained. This condition is represented in the block
diagram of Fig. 10 and can be achieved by using a level detector
with an andgate. A tripping signal is issued when the corresponding criterion
is fulfilled by one of the 2 channels. The first channel represents the instantaneous
overangle protection, where the inclination angle is greater than the first
zone setting. The second channel issues a tripping signal when the corresponding
criteria are fulfilled by both subchannels: the inclination angle should be
within the setting range (Table 7) and the current envelope
should be greater than the current envelope of the maximum load.
Table 7: 
The 95% instantaneous and timeoverangle protection settings 


Using the suggested instantaneous and timeoverangle protection overcomes the disadvantages of the definitetimeovercurrent protection and the complexity of the inversetimeovercurrent protection. In addition, faults can be sensed within the first millisecond.
CONCLUSION
The concept of the current envelope is introduced and its formula, in terms of the instantaneous phase currents, is shown. The curves of the transmission line current envelope versus fault location are obtained and analyzed. Analysis shows that the inclination angle of the current envelope can be used as a criterion for radial electrical power system protection against 3phase faults.
Based on the inclination angle an instantaneous and definitetime overangle protection system is developed. Each protection includes main zone with instantaneous action and multitimedelayed zones, which represent backup protection for the downstream transmission lines. The developed protection overcomes the disadvantages of the long timedelay of the definite timeovercurrent protection and the complexity of the inversetimeovercurrent protection of radial electrical power systems. In the developed protection system, discrimination between faults on the receiving end of the transmission line and faults on the sending end of the downstream transmission line, is achieved by the 95% instantaneous overangle protection. While the rest 5% of the transmission line length is protected by the definitetimeoverangle protection with an added time delay of 0.30.4 s.
The suggested protection system is achieved with the use of mathematical modelling of a radial electrical power system, where the model is presented as a system of differential equations for currents and a system of algebraic equations for bus voltages. These equations are written in a rectangular coordinate system, thus their number is reduced by one third. This model allows analyzing both the symmetrical transient and steadystate conditions.