Power Systems: Transmission, Faults, Stability and Protection — ESE Electrical

Weightage: Power systems is the second large block of the Electrical papers, after machines. The examiners favour per-unit conversion, fault current calculation, economic dispatch and the equal-area criterion, so a worked method for each repays the time spent.

1. System structure

Electrical energy moves through generation, transmission, sub-transmission and distribution. Power is transmitted at high voltage because for a given power the current falls in proportion to voltage and the $I^{2}R$ loss falls with its square. Indian transmission uses 765, 400, 220 and 132 kV AC, alongside HVDC links.

Load curves describe demand. The load factor is average load over peak load, the demand factor is maximum demand over connected load, and the diversity factor is the sum of individual maximum demands over the system maximum demand. A high load factor lowers the cost per unit.

2. Transmission line parameters

Line inductance per phase for a three-phase line is $L=2\times 10^{-7}\ln (GMD/GMR)$ H/m, where $GMD$ is the geometric mean distance between conductors and $GMR=0.7788r$ for a solid round conductor. Bundled conductors lower inductance and raise capacitance, and reduce corona loss and radio interference.

Line capacitance grows with the conductor spacing reduced and the height above ground. Skin effect pushes AC toward the conductor surface, raising resistance, and proximity effect adds to it.

Transposition equalises the phase inductances and capacitances and reduces interference with nearby telephone lines.

3. Line performance

Model a line by an ABCD two-port: $V_S=A{V}_R+B{I}_R$ and $I_S=C{V}_R+D{I}_R$.

For any reciprocal network, $AD-BC=1$, and for a symmetrical line $A=D$.

Voltage regulation is $(V_{NL}-V_{FL})/V_{FL}$. The Ferranti effect is a rise in receiving-end voltage above the sending end on a lightly loaded or open-circuited long line, because of its charging current. The surge impedance $Z_c=\sqrt{L/C}$ is about 400 ohms for overhead lines, and at the surge impedance loading the line neither absorbs nor supplies reactive power.

Insulators. In a suspension string the voltage is not shared equally, with the unit nearest the line taking the most. String efficiency is the voltage across the string over $n$ times that across the worst unit. A grading ring improves it.

4. Per-unit system

A quantity in per-unit is the actual value divided by its base. With base MVA and base kV:

$Z_{pu}=\frac{Z_{ohm}{\text{MVA}}_{base}}{k{V}_{base}^2}$

To change base:

$Z_{pu,new}=Z_{pu,old}\left(\frac{{\text{MVA}}_{new}}{{\text{MVA}}_{old}}\right){\left(\frac{k{V}_{old}}{k{V}_{new}}\right)}^2$

Worked example. A 30 MVA generator has $X=0.2$ pu on its own base. On a 100 MVA base, $X=0.2\times 100/30=0.667$ pu. The fault MVA at its terminals is $100/0.667=150$ MVA.

Per-unit values of a transformer are the same on both sides, which removes the need to refer quantities across voltage levels.

5. Symmetrical and unsymmetrical faults

A three-phase (symmetrical) fault is analysed with the positive-sequence network alone: $I_f=E/Z_1$.

Unsymmetrical faults use symmetrical components: positive, negative and zero sequence. For a pre-fault voltage $E$:

Worked example. With $Z_1=Z_2=j0.2$ and $Z_0=j0.1$ pu and $E=1$ pu, the three-phase fault current is 5 pu, and the single line-to-ground current is $3/0.5=6$ pu. The single-line fault exceeds the three-phase fault whenever $Z_0<Z_1$.

Zero-sequence current needs a path: it cannot flow through a delta winding from outside, and an ungrounded star blocks it. The neutral grounding method (solid, resistance, reactance, Petersen coil) sets the earth-fault current.

6. Load flow

Load flow solves the steady-state voltages from the network equations. Bus types:

• Slack bus: voltage magnitude and angle fixed, supplies the losses.
• PV (generator) bus: real power and voltage magnitude fixed.
• PQ (load) bus: real and reactive power fixed.

Gauss-Seidel is simple but converges slowly, while Newton-Raphson converges quadratically and needs few iterations whatever the system size. The fast-decoupled method treats $P$-$\delta$ and $Q$-$V$ as nearly independent. Transmission networks are highly inductive, so real power depends mainly on the angle and reactive power on the voltage magnitude.

7. Economic dispatch

Total generation cost is minimised when all units operate at the same incremental cost $\lambda =d{C}_i/d{P}_i$, ignoring losses and limits. With losses the penalty factor modifies the condition.

Worked example. Two units have incremental costs $0.2P_1+10$ and $0.4P_2+8$ and must supply 100 MW. Setting them equal with $P_2=100-P_1$: $0.2P_1+10=48-0.4P_1$, so $P_1=63.3$ MW, $P_2=36.7$ MW and $\lambda =22.7$ per MWh.

Unit commitment decides which units to run, subject to start-up cost and minimum up and down times.

8. Stability

Steady-state stability is the ability to return to a normal operating point after a small disturbance. Transient stability is the ability to remain in synchronism after a large disturbance such as a fault. The swing equation is:

$M\frac{d^2\delta }{d{t}^2}=P_m-P_e$

The equal-area criterion says a single machine against an infinite bus remains stable if the accelerating area equals the decelerating area before the angle passes its limit. The critical clearing angle and the critical clearing time are the largest values for which the fault can persist without loss of synchronism. Faster clearing, lower reactance, auto-reclosing and series compensation all improve stability.

9. Protection

Protection isolates a fault quickly and selectively.

• Relays: over-current (with a time-grading discipline), differential (compares currents in and out, used for transformers, generators and busbars), distance (measures impedance to the fault, typically with zone 1 set at about 80 percent of the line), earth-fault and Buchholz (gas-operated, for transformer internal faults).
• Circuit breakers: they must interrupt fault current by extinguishing the arc. Types include oil, air-blast, SF6 and vacuum. A breaker rating includes breaking capacity, making capacity and short-time current. The recovery voltage and its rate of rise decide arc re-ignition.
• Surge protection: the surge arrester diverts lightning overvoltage to earth, and an earth wire shields the line.

A breaker opens a faulty circuit after the relay trips it, and a fuse is a combined sensing and interrupting device for low-power circuits.

Common traps

• Mixing pu bases when elements are on different ratings.
• Treating the Ferranti effect as a loaded-line effect. It occurs when lightly loaded.
• Forgetting that zero-sequence flow needs a ground path.
• Using Gauss-Seidel iteration counts to compare with Newton-Raphson's.
• Confusing differential and distance protection.

Memory aids

• "Slack, PV, PQ": bus types.
• "Equal lambda": economic dispatch.
• "Accelerating area equals decelerating area": transient stability.

Summary

Transmission lines are modelled by ABCD parameters according to length, and per-unit analysis removes voltage-level conversions. Faults are analysed by sequence networks and give the currents that relays and breakers must handle.

Load flow, economic dispatch and stability studies keep the system running economically and in step, and protection is the last line of defence.

Exam protocol

• Convert every quantity to a common per-unit base first.
• Draw the sequence networks for unsymmetrical faults.
• Equate incremental costs, then check unit limits.
• Apply equal area to a single machine on an infinite bus.