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PHASOR DIAGRAMS II
Fault Analysis
Ron Alexander – Bonneville Power Administration
For any technician or engineer to understand the characteristics of a power system, the use of phasors and polarity are essential. They aid in the understanding and analysis of how the power system is connected and operates both during normal (balanced) conditions, as well as fault (unbalanced) conditions. Thus, as J. Lewis Blackburn of Westinghouse stated, “a sound theoretical and practical knowledge of phasors and polarity is a fundamental and valuable resource.”
A
C
B (^) Balanced System
A
C
B Unbalanced System
With the proper identification of circuits and assumed direction established in a circuit diagram (with the use of polarity), the corresponding phasor diagram can be drawn from either calculated or test data. Fortunately, most relays today along with digital fault recorders supply us with recorded quantities as seen during fault conditions. This in turn allows us to create a phasor diagram in which we can visualize how the power system was affected during a fault condition.
FAULTS
Faults are unavoidable in the operation of a power system. Faults are caused by:
- Lightning
- Insulator failure
- Equipment failure
- Trees
- Accidents
- Vandalism such as gunshots
- Fires
- Foreign material
Faults are essentially short circuits on the power system and can occur between phases and ground in virtually any combination:
- One phase to ground
- Two phases to ground
- Three phase to ground
- Phase to phase
As previously instructed by Cliff Harris of Idaho Power Company: Faults come uninvited and seldom leave voluntarily. Faults cause voltage to collapse and current to increase. Fault voltage and current magnitude depend on several factors, including source strength, location of fault, type of fault, system conditions, etc. Faults must be isolated from the power system quickly in order to minimize damage to equipment, the environment, or the power system in general, as well as to eliminate the hazard to people. Fault angle is the angle of the fault current relative to it’s respective voltage. The angle of the fault current (PF) is determined for phase faults by the nature of the source and connected circuits up to the fault location. For ground faults, you also must consider the type of system grounding.
Typical fault angles on open wire transmission lines are:
7.2-23kV = 20-45˚ lag 23-69kV = 45-75˚ lag 69-230kV = 60-80˚ lag 230kV and up = 75-85˚ lag
For faults on transmission lines, this angle is a function of the characteristics of the transmission line. High voltage transmission lines generally utilize large conductors, which characteristically have high inductance and low resistance. Thus the fault angle (current lagging voltage) will be high, usually in the range of 70-85˚. Lower voltage transmission lines usually employ smaller conductor, with higher resistance than larger conductors. Typical line fault angles are in the 40- 70˚ range. Fault resistance, especially with grounded faults, needs to also be taken into consideration too as caused by tower footings, tree limbs, ground, arc length through air, or other factors can also influence the fault angle.
Key points on making phasor diagrams from J. Lewis Blackburn’s Protective Relaying – Principles and Application, Chapter 3:
Common pictorial form for representing electrical and magnetic phasor quantities uses the Cartesian coordinates with x (the abscissa) as the axis of the real quantities and y (the ordinate) as the axis of imaginary quantities. (see figures below)
(Polar coordinates) (Rectangular coordinates)
Voltage: Voltages can be either drops or rises. Much confusion can result by not clearly indicating which is intended or by mixing the two practices in circuit diagrams. This can be avoided by standardizing on one and only one practice. As voltage drops are far more common throughout the power system, all voltages are shown and always considered to be drops from a higher voltage to a lower voltage. This convention is independent of whether the letter V or E is used for the voltage. Voltages (always drops) are indicated by either (1) a letter designation with double subscripts, or (2) a small plus (+) indicator shown at the point assumed to be at a relatively high potential. It may be helpful to consider current as a “through” quantity and voltage as an “across” quantity.
Phasor and Phase Rotation “Phasor” and “phase rotation” are two entirely different terms, although they almost look alike. AC phasors are always rotating counterclockwise at the system frequency. In contrast, phase rotation or phase sequence refers to the order in which the phasors occur as they rotate counterclockwise. The standard sequence today is a, b, c; A, B, C; 1, 2, 3; or in some areas R, S, T.
CLASSROOM LESSONS:
Let’s try applying what we have learned by drawing in the classroom a phasor diagram for each of the given fault types. We can make a diagram for pre-fault, fault, and post-fault. This is an open discussion!
SINGLE PHASE FAULTS
Single phase faults are the most common. When viewing the faulted line, the phase voltage collapses, and its current increases while also lagging the voltage by some value.
3 PHASE FAULTS
A 3 phase fault reduces all three voltages and causes a large increase and usually highly lagging fault current symmetrically in all three phases. The angle of the lag is determined by the system.
Now, let’s reverse our thinking and write in on the above graph what you think a three-phase
fault would look like.
Use these values:
Voltages shrink from 115kv to 54kV line – ground. What phase angles might they be at???
Current increases from of about 200A to what??? And what phase angles might they be at???
This is a real 3 phase fault. What do you see? Is it different than what you predicted? Why?
PHASE TO PHASE FAULTS
Phase to phase faults are sometimes the most difficult to understand. The faulted phase voltages (i.e.: b and c phases) collapse from their normal position and the phase angles can swing toward each other. During that time, current magnitudes increase as one current will lag the faulted phase to phase voltage value (Vbc) while the other phase current will be 180˚ out.
Pre-Fault Fault
Post-Fault
In assigning subscript notation, it may help to think of how you would instrument and measure the circuit with a voltmeter, ammeter, and phase angle meter. Since the inputs of your measurement instruments represent an actual resistive load, they generate the same instantaneous voltage across their terminals as the system being measured. The instantaneous direction across the phase angle meter terminals will align with the instrument polarity markings (if current and voltage leads are connected correctly). Because of this, you can verify your diagram and notation labels by visualizing how you would connect test leads to take “in-service” measurements. For voltage, you would want to visualize this as your red (±) “polarity” lead from your test set, or phase angle meter, goes to the first subscript character and black “non-polarity” lead goes to the second character. (This is analogous to a DC voltage measurement). For current, connections to the test instrument would be such that current flowed into the red (±) “polarity” lead from the first subscript notation and out the black “non-polarity” lead to the second notation character. (This is also analogous to DC as a current measurement). For current lead connections, you would be inserting a low resistance shunt between the notation subscripts. From a voltage only point of view, the measurement for current ends up looking the same as for voltage, since you have to break open the circuit and insert a burden that develops a voltage in your test instrument. An additional note: When measuring DC values with a DC meter, red to
positive and black to negative would read as a positive voltage (analogous to 0 ° AC), and a
reversed connection would read negative (analogous to 180 ° AC). For a DC meter, DC
indication is also a mathematical difference in potential (red input “minus” black input = measured voltage).
In summary, the double subscript labeling convention for voltage is voltage drop from point A to point B (VAB), and current is labeled for direction as flowing from point A to point B (IAB). For example, IAN is current flowing from point A to point N and is said to be flowing from phase A to the Neutral connection. Please note that line current is truly phase-to-neutral current for wye connected systems, and is also the current of an “equivalent wye” in delta connected systems. This is why IA can be short hand for IAN when talking line current.
Coordinate Grids
Most utilities, such as BPA, also have a convention for drawing phasors on an “x - y” coordinate grid. Except for the positive direction of reactive power, they follow standard academic convention. Even so, if you prefer to explain power with positive reactive power pointing up, as done in academic textbooks, your vector mathematics will take the form of S = V I*^ for plotting Volt-Amps. “S” is apparent power, and “∗” means to conjugate the current phasor angle before multiplying voltage and current vectors. Conjugate means to reverse the sign of the phasor angle (change to + if -, or to - if +). To eliminate the need for changing the sign of an angle, and making extra calculations, phasor drawings indicate reactive power flow as positive when pointing down.
Using this style of power flow coordinates allow the vector direction of current to describe the direction of real and reactive power flow (“+” = OUT; “-” = IN). Pointing inductive power flow down, is a quick way to show all of the needed information while making it easy to interpret in- service readings (of volts, amps, and phase angle). The quadrant your current falls in also indicates the directions of real and reactive power flow. In practice, layout and labeling on your in-service test sheets will already do this for you.
A quick comment about in-service measurements that is related to plotting phasors on coordinate grids. Try to maintain a single source as your reference for a complete set of in- service readings. This will assure that everything measured is comparable against any other measurement, since they will all have a common reference. Of course, this is not always possible, because of physical distance between measurement points (such as measurements at two different substations). In cases where different references are used, and a phasor comparison is needed between the two sets of measurements, the reference sources will need to be linked by an independent but verifiable quantity. This is typically done by selecting a common power system phase voltage, such as “A” phase, because larger utilities maintain common connections (per phase) across the entire network of 115 kV through 500 kV systems. On lower kV systems, you are not going to find yourself so lucky, so watch what you use as a measurement reference.
Plotting Phasors From In-Service Measurements
Use a similar standard convention when describing “phasor” vectors, because it will keep you from making a directional error if you stick to only one convention. Phasor notation is based on measuring system quantities at a substation and treating each measurement as if it were for a load. The connection point of your voltage is your reference, and all current measurements are taken assuming current flow away from the voltage reference point (i.e., all the lines connected to a substation bus). Because a power system is bi-directional, meaning power flow can be either forward or reverse, system current is also. You must “assume” a standard direction for all measurements, take all measurements as if voltage and current are existing in the assumed direction, and let the readings tell you which way currents are actually flowing. Current “OUT” will be in-phase with voltage and current “IN” will be 180° out-of-phase. Basically, you treat every line connected to a bus as a load, regardless of whether you know it is or isn’t.
Here is a case in point: If in-service readings were taken on both ends of a transmission line, and done correctly by two individuals, the values recorded on the in-service test forms would show both ends have per phase voltages of nearly the exact same magnitude and phase angle. The current
measurements would indicate the same magnitude but would be 180 ° out-of-phase. In other
words, if one end read 0 ° (power OUT) the other end would read 180 ° (power IN) on their phase
angle meter.
When you are taking in-service readings for substation equipment with multiple system connections, such as transformers and bus differential schemes, you take all current readings assuming current flow towards or “into” the device. It matters not that it has only two connections or several, because visualizing the assumed current direction is really no different
+jX
+R
+Y
+X +P
Impedance Current & Voltage Power
+Q
Real Power P = VI cos θ, units Watts
Reactive Power Q = VI sinθ, units Var (VA reactive) Lagging load (positive vars) Leading load (negative vars)
Pythagorean Theorem: c² = a² + b²
Vectors may be expressed either in rectangular or polar coordinates.
Rectangular Polar R + j X Z ∠∠∠∠θ°°°° R = Z cos θ Z = √(R^2 + X^2 ) j X = j Z sin θ θ = arc tan (X/R) or tan-1 (X/R)
The equivalent value of the operator j is √-1. Therefore, assuming a vector of +1∠ 0 ° or unity at zero angle, multiplying: by j rotates it counter clockwise to a position of 1∠ 90 °, by j^2 = √-1 x √-1 = -1 rotates from original position to 1∠ 180 °, and j^3 = -1 x j or 270° from the original position.
Rectangular Coordinate Math:
Adding & Subtracting (easy) Multiplication (hard) Division (hardest)
R
jXL
Z
θ
( 30 + j 20 ) +( 20 +j10) = 30 + 20 + j 20 + j 10 = 50 + j 30
( 30 + j 20 ) - ( 20 +j10) = 30 − 20 + j 20 − j 10 = 10 + j 10
Examples of math using rectangular coordinates : Addition: Subtraction: Multiplication: 4 + j 2 4 + j 2 4 + j 2 -2 - j 3 -2 - j 3 -2 - j 3 2 - j 1 6 + j 5 - j 12 - j^26 -8 - j 4 -8 - j 16 - j^26 (since j^2 = -1) -2 - j 16 is the product.
Polar Coordinate Math:
Multiplication (easy) Division (easy)
Examples of math converting rectangular to polar coordinates:
4 + j 2 = A∠θ 1 -2 - j 3 = B∠θ 2
A = √(4^2 +2^2 ) = √20 = 4.48 B = √(-2^2 +-3^2 ) = √13 = 3. θ 1 = arc tan (2/4) = arc tan 0.5 = 26.5° θ 2 = arc tan (-3/-2) = arc tan 1.5 = 56.3° A = 4.48∠26.5° B = 3.61∠(56.3° + 180°)
Alternate example of polar to rectangular, and rectangular to polar conversion (referencing graph below):
P ⇒ R x = r cosθ y = r sinθ
R ⇒ P
θ = tan-1( y/x )
10 ∠ 20 °x 20 ∠ 10 °=(10x20)∠( 20 °+ 10 °)= 200 ∠ 30 °
Three phase fault (on the 500kV system, C phase opens late):
Phase to phase fault (b-c phase):
Never to be overlooked are the acknowledgements and assistance by:
- Protective Relaying Principles and Applications by J. Lewis Blackburn
- Bonneville Power Administration System Protection and Control Branch
- Steve Laslo, BPA Technical Training
- Richard Becker, BPA Electrical Engineer
- Cliff Harris, Idaho Power Company
