(a) Discuss the percentage differential Relay with harmonic restraint with the help of diagram and also draw the conceptual representation of it. (20 marks)

(b) A sinusoidal voltage of 10 V amplitude at 100 Hz is applied to a lead network shown in f

Question

(a) Discuss the percentage differential Relay with harmonic restraint with the help of diagram and also draw the conceptual representation of it. (20 marks)

(b) A sinusoidal voltage of 10 V amplitude at 100 Hz is applied to a lead network shown in figure. The phase difference between the input voltage $V_i(t)$ and output voltage $V_o(t)$ is 44.43°. If $C = 0.1 \mu F$ and $R_1 = 100 k\Omega$, determine the value of $R_2$ and the magnitude of steady state output voltage. (20 marks)

(c) Consider a connected graph G = (N, A) with N nodes and A arcs, and a weight ωij for each arc (i, j)∈A.

(i) Define minimum weight spanning tree (MST).

(ii) If all arc weights of G are distinct, prove that there exists a unique MST. (10 marks)

UPSC Answer Check · Accepted Answer

Begin with a brief introduction on protection systems, then allocate approximately 40% effort to part (a) on percentage differential relays with harmonic restraint, 35% to part (b) on the lead network numerical solution, and 25% to part (c) on MST definition and uniqueness proof. Structure each part with clear headings, present derivations stepwise, and conclude with practical significance of harmonic restraint in transformer protection and MST applications in power system planning.
- Part (a): Principle of percentage differential protection with percentage slope characteristic; need for harmonic restraint (2nd harmonic) to prevent maloperation during transformer inrush; block diagram showing CTs, restraint and operating coils, harmonic filter circuit, and trip logic
- Part (a): Conceptual representation showing percentage differential characteristic with slope, operating region, and restraint region; explanation of why 15-20% slope is typical for percentage differential
- Part (b): Correct transfer function derivation for lead network; phase angle condition tan(φ) = (R1+R2)/(ωCR1R2) or equivalent; solving for R2 using given phase difference of 44.43°
- Part (b): Magnitude calculation |Vo/Vi| = ωCR2/√[1+(ωCR2)²] or appropriate expression; final numerical values with proper units (R2 in kΩ, |Vo| in volts)
- Part (c)(i): Formal definition of MST as a spanning tree with minimum total weight; connected, acyclic subgraph containing all nodes with N-1 arcs
- Part (c)(ii): Proof of uniqueness using cut property or cycle property; argument that with distinct weights, any two MSTs would lead to contradiction via edge exchange
- Application context: Mention use of harmonic restraint in Indian power transformers (NTPC, state electricity boards) and MST in optimal transmission network design

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	22%	11	Accurately explains percentage differential relay principle with correct slope characteristic, identifies 2nd harmonic as inrush restraint component, correctly defines MST with spanning tree properties, and presents valid uniqueness proof using cut/cycle property with distinct weight assumption	Basic understanding of differential protection and MST definition but confuses harmonic restraint with other blocking methods, or presents incomplete/circular proof for uniqueness	Fundamental errors such as describing overcurrent instead of differential relay, incorrect harmonic order (e.g., 3rd instead of 2nd), or defining MST as shortest path instead of spanning tree
Numerical accuracy	18%	9	Correctly calculates R2 ≈ 50 kΩ (acceptable range 48-52 kΩ) and \|Vo\| ≈ 7.07 V using proper complex impedance analysis; shows intermediate values with correct unit handling throughout	Correct method but arithmetic errors leading to R2 within 10% of correct value, or correct R2 but magnitude calculation error due to phase angle misuse	Major errors in transfer function, incorrect phase angle formula, or order-of-magnitude errors in final values; missing units or incorrect unit conversions
Diagram quality	16%	8	Clear labeled diagram for part (a) showing CT connections, restraint coil, operating coil, harmonic filter block, and trip circuit; conceptual representation shows percentage slope characteristic with operating/restraint regions properly demarcated	Diagrams present but missing key labels (e.g., filter block) or conceptual representation lacks clear slope indication; circuit topology correct but poorly organized	Missing diagrams, incorrect circuit topology (e.g., lag network instead of lead), or illegible sketches; no conceptual representation of percentage differential characteristic
Step-by-step derivation	22%	11	Systematic derivation: for (a) shows how percentage restraint equation I_op > K·I_res + I_pickup prevents maloperation; for (b) derives transfer function Vo/Vi = jωCR2/(1+jωCR2) with clear impedance steps; for (c)(ii) uses exchange argument or cut property rigorously	Derivations present but skips critical steps (e.g., jumps from phase angle to R2 value) or presents proof by example rather than general argument for MST uniqueness	No derivations shown, only final answers stated, or logical gaps that invalidate proofs; confuses derivation with description
Practical interpretation	22%	11	Explains why harmonic restraint is critical for EHV transformer protection in Indian grid (400kV/765kV), discusses CT saturation effects on slope setting; relates lead network to phase compensation in relaying; connects MST to minimum cost transmission expansion in state power grids	Generic mention of transformer protection applications without specific context, or superficial connection of MST to network design without elaboration	No practical context provided, or irrelevant applications cited; fails to explain why percentage differential is preferred over simple differential

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