Q1Protection of Power System

Power system faults are broadly classified into symmetrical (balanced) faults and unsymmetrical (unbalanced) faults. The only symmetrical fault is the three-phase fault (with or without simultaneous ground contact), in which all three phases are shorted together, preserving the balanced, symmetrical nature of the resulting fault currents in all three phases; although statistically the rarest type of fault, the three-phase fault is analytically important because it produces the severest fault current magnitude and is used as the reference case for circuit breaker and switchgear rating calculations. Unsymmetrical faults, which constitute the large majority of faults actually occurring on a power system, include the single line-to-ground (LG) fault, the line-to-line (LL) fault, and the double line-to-ground (LLG) fault, each of which disturbs the three-phase symmetry of the system and requires symmetrical component analysis (decomposing the unbalanced fault condition into positive-, negative-, and zero-sequence balanced component networks) rather than the simpler balanced per-phase analysis sufficient for the three-phase fault.

Methods of Calculating Short Circuit Fault Currents

For a three-phase symmetrical fault, the fault current is calculated directly using the single-phase (positive-sequence) equivalent circuit of the system, since all three phases behave identically; the fault current is simply the pre-fault source voltage divided by the total positive-sequence impedance from the source to the fault point, Ifault = V/Z1, with per-unit system representation and the reactance (or impedance) diagram of the network commonly used to simplify calculation across multiple interconnected generators, transformers, and lines.

For unsymmetrical faults, the method of symmetrical components is used, in which the actual unbalanced three-phase fault condition is represented as the superposition of three separate, mutually independent balanced sequence networks: the positive-sequence network (identical to the normal balanced operating network), the negative-sequence network (structurally similar to the positive-sequence network but typically using slightly different reactance values, since rotating machines present a different reactance to negative-sequence currents than to positive-sequence currents), and the zero-sequence network (which additionally depends critically on transformer winding connections and system grounding arrangements, since zero-sequence current requires a return path through ground or a neutral connection). For a single line-to-ground fault, the three sequence networks are connected in series at the fault point, and the fault current is calculated as three times the resulting common sequence current, since the actual fault current in the faulted phase is the sum of the equal positive-, negative-, and zero-sequence fault currents: If = 3*E/(Z1+Z2+Z0), where E is the pre-fault phase voltage and Z1, Z2, Z0 are the total positive-, negative-, and zero-sequence impedances viewed from the fault point.

For a line-to-line fault, only the positive- and negative-sequence networks are involved (connected in parallel at the fault point, with the zero-sequence network left unconnected since no ground path is involved), giving a fault current of If = E*sqrt(3)/(Z1+Z2). For a double line-to-ground fault, all three sequence networks are again involved, but connected in a parallel combination (positive-sequence in series with the parallel combination of negative- and zero-sequence networks) rather than the simple series connection used for the single line-to-ground case, giving a somewhat more involved fault current expression that nonetheless follows directly from applying Kirchhoff's laws to the resulting sequence network interconnection. In each of these unsymmetrical fault cases, the underlying principle is the same: the boundary conditions specific to that particular fault type (which phases are shorted, whether ground is involved) dictate a specific interconnection pattern among the three sequence networks, and solving the resulting combined network yields the sequence currents, which are then transformed back into actual phase currents using the standard symmetrical component transformation.

Current Transformers (CTs)

A current transformer is a specialized instrument transformer whose primary winding (typically consisting of just a single turn, formed by the power system conductor itself passing through the CT core, or a few turns for lower primary current ratings) is connected in series with the power circuit whose current is to be measured or monitored, and whose secondary winding, having a much larger number of turns, delivers a proportionally reduced, standardized secondary current (commonly rated at 1A or 5A at rated primary current) to connected metering instruments or protective relays, while simultaneously providing electrical isolation between the high-voltage power circuit and the low-voltage secondary metering and protection equipment.

Ratio Error and Phase Angle Error

In an ideal current transformer, the secondary current would be exactly equal to the primary current divided by the turns ratio, and would be exactly 180 degrees out of phase with the primary current (accounting for the standard dot convention). In practice, however, a real CT draws a small magnetizing (exciting) current to establish the core flux needed to induce the secondary EMF, and this magnetizing current is drawn from the primary ampere-turns, meaning only the remaining portion of the primary ampere-turns is actually available to balance the secondary ampere-turns; this causes both a ratio error (the actual transformation ratio deviates from the nominal turns ratio) and a phase angle error (the secondary current is not exactly 180 degrees out of phase with the primary current, due to the magnetizing current having both an in-phase, core-loss component and a quadrature, magnetizing component relative to the flux). The ratio error is defined as the percentage difference between the nominal transformation ratio and the actual transformation ratio, expressed as a percentage of the actual primary current, while the phase angle error is defined as the angular displacement (typically expressed in minutes of arc) between the reversed secondary current phasor and the primary current phasor, ideally zero in an ideal transformer.

where Kn is the nominal (turns) ratio, Is is the actual secondary current, and Ip is the actual primary current.

Reasons for CT Errors

• Magnetizing (exciting) current: the fundamental root cause of both ratio and phase angle error, since any current diverted to magnetize the core is current that cannot contribute to balancing the secondary ampere-turns, and this exciting current itself has both in-phase (core loss) and quadrature (magnetizing) components relative to the working flux.
• Core material and construction: cores with lower permeability or higher hysteresis and eddy current losses require a proportionally larger magnetizing current for a given flux level, worsening both ratio and phase angle error; this is why CTs intended for high-accuracy metering or sensitive protection applications use high-permeability core materials such as nickel-iron alloys.
• Burden (the total impedance connected to the secondary winding, comprising the connected relays, meters, and the resistance of the interconnecting leads): a higher secondary burden requires a correspondingly higher secondary EMF (and hence higher flux and higher magnetizing current) to drive the required secondary current through that burden, worsening both ratio and phase angle error as burden increases.
• Primary current magnitude relative to CT rating: errors are generally worst at very low primary currents (where the magnetizing current, though small in absolute terms, represents a proportionally larger fraction of the small primary current) and can also increase significantly at primary currents well above the CT's rated value, particularly as the core approaches saturation.

Remedies for CT Errors

• Using a high-permeability, low-loss core material (such as grain-oriented silicon steel for general-purpose applications, or nickel-iron alloys for the most demanding, high-accuracy metering CTs), which reduces the magnetizing current required for a given flux level and hence directly reduces both ratio and phase angle errors.
• Keeping the connected secondary burden as low as practical (within the CT's rated burden capability), since a lower burden reduces the required secondary EMF and hence the required magnetizing current.
• Using turns compensation, in which the actual secondary turns count is deliberately made slightly different from the exact nominal turns ratio, in a direction chosen to compensate for the anticipated ratio error at the CT's expected operating burden and current, effectively canceling out much of the ratio error at the design operating point.
• Selecting a CT with an appropriately high accuracy class and burden rating for the specific application, ensuring the CT is neither under-utilized (operating far below its rated burden, which is generally acceptable) nor over-burdened (operating with a secondary burden exceeding its rated capability, which significantly worsens both types of error and, for protection-class CTs, risks core saturation during heavy fault currents that could compromise correct relay operation during the very fault conditions the protection scheme is intended to detect).