৪৯তম বিসিএস ⎯ তথ্য ও যোগাযোগ প্রযুক্তি (EEE) [ ৮৯২]

Types of materials used for insulators
Insulators for overhead lines must have:

1. High electrical resistance
2. Mechanical strength
3. Weather resistance
4. The commonly used materials are:

Porcelain: Most widely used for high voltage transmission lines.
Advantages: strong, durable, resistant to weather and pollution.
Glass:Used occasionally, especially for lower voltages.
Advantages: good dielectric strength, visible cracks.
Disadvantages: brittle, more prone to breakage.
Steatite:A type of ceramic.
Used in some special cases, but not as common as porcelain.

Most commonly used material, among all these, porcelain is the most common material for overhead line insulators.

String efficiency (η\etaη) of a suspension insulator string is defined as:

η=Withstand voltage of the string/Sum of individual insulator voltages×100%
It measures how effectively the applied voltage is shared among all the insulator units in the string.
Due to shunt capacitance and leakage, the voltage across each insulator may not be equal, so the string efficiency is less than 100%.

In a D.C. system, there is no capacitive effect. All insulators share voltage almost equally because there is no leakage current due to capacitance.
⇒String efficiency for D.C. system is nearly 100%

1. Tension in the conductor
Sag is inversely proportional to the tension in the conductor.
Formula (for a uniform conductor on a level span):
S=ωL2/8T

S is the sag,
ω is the weight per unit length of the conductor,

L is the span (distance between the supports),

T is the horizontal tension in the conductor.

2. Conductor material
The weight of the conductor depends on its material.
Heavier materials (like copper or aluminum) cause more sag under the same tension and span.
Also, thermal expansion and elasticity of the material influence sag.
 
3. Height of tower
Indirectly, the height of the tower sets the allowable sag. Taller towers allow more sag while maintaining safety clearance.
So while the height doesn’t directly affect sag physically, it constrains the design and influences how sag is maintained.

In a three-phase system, the phase voltage VϕV_\phiVϕ​ is related to the line voltage VLV_LVL​ as:

V_ϕ=V_L/√3 =33/√3 =19.05

When the span of an overhead line conductor increases, the sag (the vertical displacement of the conductor from a straight line) typically increases. This is because the sag in a conductor is influenced by several factors, with the span length being one of the most significant.

The sag in a conductor is caused by the weight of the conductor, which acts vertically downward due to gravity. The conductor is supported at two points, and due to its own weight, the tension created in the conductor stretches it between these supports, causing it to sag.

Relationship Between Span and Sag:
The general formula for sag S in an overhead conductor is given by:
S=ωL²/8T

S is the sag,
ω is the weight per unit length of the conductor,

L is the span (distance between the supports),

T is the horizontal tension in the conductor.

From this formula, it's clear that sag is directly proportional to the square of the span length. This means that as the span L increases, the sag  S increases significantly, provided other factors like tension and weight remain constant.

Impact of Increased Span:
As the span increases, the force exerted by the weight of the conductor stretches it further, leading to more sag. The conductor is unable to maintain the same tension over a longer span because the gravitational pull on the conductor’s mass becomes more significant over the greater distance.

Practical Considerations:
In practical applications, increasing the span also affects the tension in the conductor. If the span becomes too long, it can lead to excessive sag, which may result in the conductor coming too close to the ground, causing safety hazards or even contact with trees or buildings. To manage sag over long spans, engineers often increase the tension in the conductor or use stronger materials to withstand the increased forces.

Working tension,T = working stres * Area of Conductor = 1050 *3.2 =3360 kg

To find the wind load per meter length of the conductor, we use the formula for wind load:

Wind Load = (Wind Pressure * Projected Area) / Span Length

Given:
- Wind Pressure = 30 kg/m²
- Projected Area = Diameter of conductor × Span length = 20 mm × 160 m
- Span Length = 160 m

First, we calculate the projected area:

Projected Area = Diameter × Span length = 20 mm = 0.02 m
Projected Area = 0.02 m × 160 m = 3.2 m²

Now we can calculate the wind load per meter length:

Wind Load = (30 kg/m² * 3.2 m²) / 160 m = 0.6 kg/m

Thus, the correct answer is  0.6 kg/m.

String efficiency in suspension insulators is defined as the ratio of the total voltage applied across the insulator string to the total voltage across the individual insulator units. The efficiency improves when the voltage is evenly distributed across all the insulator units, which reduces the potential for partial discharges and overloading of individual insulators.

The voltage across each unit in the string depends on the shunt capacitance of the insulators. The presence of shunt capacitance causes a voltage drop across the insulator string, leading to an unequal voltage distribution, which is detrimental to the overall efficiency.

The voltage distribution becomes more uniform across the insulators, meaning each insulator will experience a more balanced voltage.This results in higher string efficiency because the insulators are subjected to more even voltage stress, improving their overall performance and reducing the likelihood of breakdowns.

When the temperature of an overhead line increases, it leads to thermal expansion of the conductor material. This expansion affects both the stress on the line and its length.

Increase in temperature causes the conductor to expand, which means the length of the line will increase.

As the conductor expands and its length increases, the tension (or stress) in the conductor decreases because the line can elongate to accommodate the temperature change, reducing the overall strain on the system.

The corona effect is influenced by several factors, including:

Condition of the atmosphere: The atmospheric pressure, temperature, humidity, and pollution levels directly affect the breakdown voltage and the electric field around the conductor. Higher humidity reduces the ionization threshold for corona, while lower air pressure (such as at higher altitudes) can make it easier for the corona effect to occur. Pollution can also cause corona to occur at lower voltages due to the presence of particles in the air that act as ionization centers.
Size and spacing of conductors: Larger conductors reduce the intensity of the electric field around the conductor, lowering the likelihood of corona discharge. Larger conductor size increases the surface area over which the voltage is distributed, reducing the electric field intensity.

The spacing between conductors also plays a role. Closer spacing between conductors increases the electric field and can result in a higher potential for flashover and corona discharge. Wider spacing helps reduce this effect by reducing the electric field intensity between conductors.

Line voltage: Higher line voltages directly increase the electric field around the conductors. If the voltage exceeds the critical disruptive voltage, it leads to ionization of the air and the onset of corona discharge. Therefore, higher voltage levels are more likely to cause corona.

The corona effect occurs when the voltage applied to a conductor is high enough to ionize the surrounding air, causing partial electrical breakdown. This leads to the formation of a visible, glowing discharge around the conductor, which can cause power loss, radio interference, and degradation of the conductor material.

To reduce the corona effect, we typically aim to reduce the electric field around the conductor. This can be achieved by various methods, such as changing the conductor size or spacing.
Increasing the size of the conductor increases the conductor's surface area, which reduces the electric field intensity around it. A larger conductor means that the voltage is spread over a larger area, thereby reducing the likelihood of corona discharge.This would increase the electric field around the conductor, which would make the corona effect worse, not better. A smaller gap between the conductors increases the chance of a flashover or arc discharge, rather than reducing the corona effect.

The minimum phase-neutral voltage at which corona discharge occurs is referred to as the critical disruptive voltage.

Explanation:
Corona discharge occurs when the electric field around a conductor becomes strong enough to ionize the surrounding air, causing a discharge. This usually happens at higher voltages and is a form of partial electrical breakdown.
The critical disruptive voltage is the minimum voltage at which this corona discharge happens. At this voltage, the air around the conductor starts to ionize and conduct, leading to a corona.

Why the other options are incorrect:
(খ) Visual critical voltage: This term refers to the minimum voltage at which the corona becomes visible as a faint glow around the conductor. It's related but not the same as the disruptive voltage.
(গ) Flash-over voltage: This is the voltage at which arc-over occurs, not corona. Flash-over refers to a complete breakdown of the insulating medium (such as air or other dielectric materials), whereas corona is a partial discharge.
(ঘ) None of the above: This is incorrect because critical disruptive voltage is the correct answer.

Critical Clearing Time in Power Systems

Definition: The term critical clearing time refers to the maximum time allowed for the circuit breakers to clear a fault in a power system without causing system instability. It is a key parameter in transient stability analysis, as it determines the stability of the power system following a fault.

Transient Stability: Transient stability in a power system refers to the ability of the system to maintain synchronism when subjected to a transient disturbance, such as a short circuit, sudden loss of generation, or a sudden change in load. The system is said to be transiently stable if all machines in the system remain in synchronism after the fault is cleared.

Role of Critical Clearing Time: When a fault occurs, it causes a sudden change in the power angle of the synchronous machines in the system. If the fault persists for too long, the power angle may exceed its stability limit, causing the machines to lose synchronism. The critical clearing time is the maximum duration for which a fault can exist without causing instability. If the fault is not cleared within this time, the system becomes unstable.

Factors Affecting Critical Clearing Time:

Fault Location: Faults closer to the generators have a more significant impact on the stability of the system, reducing the critical clearing time.

Pre-fault Operating Conditions: High power transfers or heavily loaded systems have lower critical clearing times.

System Configuration: The topology and connectivity of the power system, such as the number of parallel transmission lines, influence the critical clearing time. Fault Impedance: High impedance faults are less severe and result in longer critical clearing times compared to low impedance faults.

Importance of Critical Clearing Time:

Ensures the stability of the power system during transient disturbances. Helps in designing protective relays and circuit breakers to operate within the permissible time limits. Provides insights for planning and maintaining power system reliability.

Willaim D. Stevenson, Elements of Power System Analysis, Fouth Edition, McGraw-Hill
International Editions, Civil Engineering Series, McGraw-Hill Inc.

The swing equation is given as:

J x (d2 (δ)/dt²) = Tm - Te

Where:

J = Moment of inertia of the rotor (kg·m²).

delta = Rotor angle or power angle (radians).

d2(δ)/dt² = Angular acceleration of the rotor (rad/s²).

Tm = Mechanical torque supplied by the prime mover (Nm).

Te = Electromagnetic torque developed by the synchronous machine (Nm).

Conceptual Understanding:

The swing equation models the dynamics of the rotor of a synchronous machine by quantifying the net torque acting on the rotor. The mechanical torque (Tm) is provided by the prime mover, while the electromagnetic torque (Te) opposes it. The difference between these two torques results in either acceleration or deceleration of the rotor, leading to changes in the rotor angle (δ).

Importance:

Used to analyze transient stability of synchronous machines in power systems.

Describes the rotor's dynamic response during disturbances such as faults or load changes.

Helps in designing and optimizing control systems for power system stability.

Equal Area Criterion in Power-Angle Curve
Definition: The equal area criterion is a method used to analyze the stability of a synchronous machine under transient conditions. It is based on the principle that the area of acceleration (A₁) and the area of deceleration (A₂) in the power-angle curve should be equal for the system to remain stable after a disturbance. The power-angle curve represents the relationship between the electrical power output of a synchronous machine and the rotor angle (δ).
Working Principle: When a disturbance occurs, the rotor angle changes, causing a temporary imbalance between the mechanical input power (Pm) and the electrical output power (Pe). The rotor accelerates or decelerates depending on whether Pm is greater than or less than Pe. The equal area criterion ensures that the energy gained during acceleration is equal to the energy lost during deceleration, allowing the rotor to settle at a new equilibrium point.
A₁=∫(Pm-Pe)dδ =0
This option accurately defines the area of acceleration (A₁) in the power-angle curve. The integral represents the energy gained by the rotor during acceleration, which occurs when the mechanical input power (Pm) exceeds the electrical output power (Pe). The limits of integration are 8, (the initial rotor angle at the start of the disturbance) and 8c (the critical rotor angle where the system transitions to deceleration). The condition for stability is that the total energy gain (A₁) during acceleration equals the total energy loss (A2) during deceleration.

The power transfer in long transmission lines is limited by the magnitude of the voltages at the two ends, the reactance between the two ends, and the sine of the angle between the two voltages.

Steady-state stability limit is given by P =( EV/X )* sinδ

The power transfer capability of a transmission line is the most affected by Inductance.

Important:

The power transfer capability of an AC transmission line is influenced by the stability limit; The power transfer capability of the existing transmission system can be enhanced through the improvement of stability limits.

we can improve steady-state stability limit by increasing transmission voltage (V) or by decreasing line reactance (X)

Transient stability refers to the ability of a power system to maintain synchronism after being subjected to a major disturbance, such as a system fault (e.g., short circuit), a sudden loss of generation, or a sudden load change. It focuses on the short-term dynamic response of the system, typically within seconds to minutes, immediately after the disturbance occurs.

When a fault or large disturbance occurs, the power system experiences a temporary imbalance between mechanical power (generated by turbines) and electrical power (supplied to the grid). This imbalance can cause the rotor angles of synchronous machines to deviate. The system must then recover and return to steady-state conditions without losing synchronism—i.e., all generators must continue to operate in harmony with each other.
 Transient stability primarily deals with the system's ability to recover from major disturbances like faults, loss of generation, or large load changes. These disturbances are significant enough to cause a temporary imbalance, and transient stability studies assess whether the system can remain synchronized after these events.

In transient stability studies, the primary goal is to analyze how a power system responds to a major disturbance (such as a fault or sudden change in load) and to determine if the system remains synchronized or if it loses synchronism.

When a fault occurs, the system’s stability may be temporarily disrupted. The transient stability study evaluates whether the system can recover from that disturbance and return to its normal operating conditions without losing synchronism among its generators.

Let's break down the options:

ক) Monitoring system faults:While fault monitoring is important in power systems, transient stability specifically focuses on the system's response after a fault occurs, particularly whether it can remain synchronized after the fault is cleared. Monitoring faults alone doesn't fully capture the focus of transient stability.

খ) Determining whether the system remains in synchronism after a disturbance:Correct answer. The main purpose of transient stability studies is to determine if the system can remain in synchronism (i.e., if the generators continue to operate together without losing synchronization) after a major disturbance such as a fault or sudden load change.

গ) Predicting long-term load behavior:Incorrect. Long-term load behavior and its effects are typically part of steady-state stability analysis, not transient stability. Transient stability is concerned with short-term response to disturbances, not long-term predictions.

ঘ) Estimating power generation levels: Incorrect. Estimating power generation levels is part of load flow or power generation analysis, not specifically related to transient stability. Transient stability is more concerned with the dynamic response of the system following disturbances.

The first-swing stability problem is a key concept in transient stability analysis. It focuses on the short-term behavior of the power system right after a disturbance, typically a fault, and the system's ability to maintain synchronism during that initial period.
First-Swing Stability: First-swing stability deals with the system’s ability to remain in synchronism with the rest of the system during the first few moments (typically within the first second) after a major disturbance, such as a fault or a sudden change in load. In this analysis, we examine whether the rotor angle of the synchronous machine will return to its steady state or continue to deviate, potentially leading to loss of synchronism. The analysis does not extend over a long period of time or involve repeated faults—it specifically looks at the system's initial response to the disturbance.(Willaim D. Stevenson, Elements of Power System Analysis, Fouth Edition, McGraw-Hill
International Editions, Civil Engineering Series, McGraw-Hill Inc.)

Dynamic stability and steady-state stability are both concepts used in power system stability analysis, but they differ in the types of disturbances and behaviors they address.

Dynamic Stability:
1. Dynamic stability refers to the ability of a power system to maintain stable operation after being subjected to large disturbances, such as system faults, sudden load changes, or sudden loss of generation.
2. These disturbances can cause significant changes in the system’s operating conditions, and dynamic stability analyzes how well the system can recover from these large disturbances over time.
3. Dynamic stability often requires solving the swing equation and considering the electromechanical dynamics of the system (including the inertia of the rotating machinery and the damping effects).

Steady-State Stability:
1. Steady-state stability, on the other hand, deals with the system’s ability to maintain stable operation under small and gradual disturbances, such as minor fluctuations in load or slight variations in generation.
2. In steady-state stability analysis, the system’s operating conditions do not change significantly, and the primary concern is ensuring that the voltage and frequency remain within acceptable limits.
3. It is more concerned with long-term, gradual variations and ensures that these variations do not lead to large deviations from steady operation.

The Equal-Area Criterion is a method used in power system stability analysis, particularly for single-machine systems. It helps to determine whether a system will remain stable after a disturbance. The basic principle behind the criterion is that the area under the power-angle curve during the acceleration period must be equal to the area under the curve during the deceleration period for the system to remain stable. Acceleration Area: This is the area under the power-angle curve when the mechanical power (Pm) exceeds the electrical power (Pe), causing the rotor to accelerate.
Deceleration Area: This is the area under the curve when the electrical power (Pe) exceeds the mechanical power (Pm), causing the rotor to decelerate.
The Equal-Area Criterion states that the system remains stable as long as the accelerating area equals the decelerating area.
If the accelerating area is larger than the decelerating area, it means that the rotor is accelerating for a longer period than it is decelerating, which results in an unstable condition.
In this case, the system will lose synchronism, meaning the rotor will not be able to return to its steady state, and the system will become unstable.

In transient stability studies, a simplified model of the synchronous machine is often used to reduce complexity and focus on the core dynamics of rotor behavior and stability analysis. This simplified model makes several assumptions that help streamline the calculations.

Let's go through the assumptions:

ক) Balanced three-phase positive-sequence conditions:

Correct assumption. A synchronous machine typically operates under balanced three-phase conditions, meaning the three phases are equal in magnitude, displaced by 120° in phase, and have a positive sequence (a standard configuration for stable operation).

খ) Machine excitation is constant:

Correct assumption. In many simplified models, the excitation (the control of the magnetic field in the rotor) is assumed to be constant. This simplifies the analysis because it avoids having to model the dynamic response of the excitation system.

গ) The machine operates with minimal electrical losses:

Correct assumption. The simplified model often assumes that electrical losses (such as core losses and copper losses) are negligible. This assumption is used to make the model more straightforward and focus on the key variables that affect stability, such as mechanical power and electrical power.

ঘ) The machine’s losses, saturation, and saliency are considered:

Incorrect assumption. In a simplified model for transient stability, losses, saturation, and saliency (the non-symmetric distribution of the magnetic field in the rotor) are neglected. Including these factors would significantly complicate the analysis without providing substantial benefits for most transient stability studies. Therefore, the simplified model does not consider these elements to keep the calculations manageable.

 Steady-state stability involves the system's ability to maintain normal operating conditions with gradual changes.

In the swing equation, the "m" in Pm​ and Pe​ stands for mechanical and electrical power, respectively. These terms describe the torque produced by the generator, with Pm​ being the mechanical power (from the prime mover) and Pe​ being the electrical power (output of the generator).

The swing equation describes how the rotor angle δ\deltaδ of a synchronous machine changes with time in response to changes in mechanical and electrical power. It governs the dynamics of the system, particularly the transient stability.

Form of the Swing Equation:
The correct form of the swing equation in the per-unit system is:
dδ/dt=(Pe−Pm)/H

where:
dδ/dt is the rate of change of the rotor angle δ\deltaδ with respect to time.
Pm​ is the mechanical power supplied by the prime mover (the input power).
Pe​ is the electrical power produced by the generator (the output power).
H is the inertia constant, which represents the machine's ability to resist changes in speed (stored energy in the rotor at synchronous speed).
Pm (Mechanical Power): This is the power supplied to the generator by the prime mover (such as a steam turbine or hydro turbine).Pm​ is the input mechanical power that causes the rotor to turn.
Pe (Electrical Power):This is the power produced by the synchronous generator and supplied to the grid. Pe​ is the output electrical power that is being delivered to the load or grid.

Why the Swing Equation is Important:
The swing equation describes how the rotor angle δ\deltaδ will evolve over time as a function of the difference between mechanical power and electrical power.
If Pm>PeP_m > P_ePm​>Pe​, the rotor accelerates, and δ\deltaδ increases.
If Pe>PmP_e > P_mPe​>Pm​, the rotor decelerates, and δ\deltaδ decreases.
The system remains stable if the rotor angle does not exceed a critical value, called the critical clearing angle, after a disturbance. If the angle exceeds this value, the system loses synchronism and becomes unstable.

The Swing Equation for Coherent Machine
Machines which are swing together are called 
In a stability study for a large system with many machines geographically dispersed over a wide
area it is desirable to minimize the number of swing equations to be solved. This can be done if the
transmission - line fault, or other disturbance on the system, affects the machines within a power
plant so that their rotors swing together. I n such cases the machines with in the plant can be
combined into a single equivalent machine just as if their rotors were mechanically coupled and
only one swing equation must be written for them. Consider a power plant with two generators
connected to the same bus which is electrically remote from the network disturbances. The swing
equations on the common system base are:

The critical clearing angle is a key concept in power system stability analysis, especially when dealing with faults or disturbances in the system. It represents the maximum allowable angle between the rotor and the synchronous machine's reference axis, beyond which the system will lose synchronism and become unstable.

Let's break it down:

ক) The angle at which mechanical power exceeds electrical power: This is not correct. The critical clearing angle isn't defined by the point where mechanical power exceeds electrical power. It's related to system stability following a disturbance.

খ) The maximum power the system can deliver: This refers to the maximum power transfer capability of the system, but it is not the critical clearing angle. The critical clearing angle is about the system's ability to return to synchronism after a disturbance, not the maximum power it can transfer.

গ) The angle beyond which the system becomes unstable: Correct answer. The critical clearing angle is the maximum allowable angle before instability occurs. If the power angle (the angle between the rotor and the reference axis) exceeds this value, the system loses synchronism and becomes unstable. In other words, the system will not be able to return to a stable operating condition after the fault is cleared.

ঘ) The initial rotor angle at steady-state: This refers to the angle when the system is operating normally (at steady state) before any disturbances occur. It is not the critical clearing angle, which comes into play only after a disturbance or fault. John J. Grainger, William D. Steevnson, Jr., Power System Analysis, McGraw-Hill Series in
Electrical and Conputer Engineering, McGraw-Hill Inc.

The per-unit system is a method of normalizing electrical quantities to a common base, making it easier to work with different voltage, current, and power levels across various parts of the power system. In the context of stability analysis, the per-unit system helps by simplifying the equations and reducing the complexity of calculations.

Normalization of Values: The per-unit system expresses values as a ratio to a defined base value (such as the base voltage, base power, or base current). For example, if the base power is 100 MVA, a generator producing 50 MW is expressed as 0.5 p.u. (per unit) on that base, meaning it is 50% of the base power. This normalization reduces the need to deal with large variations in units (like volts, amperes, or watts) across different components of the system, allowing easier comparison and analysis.

Simplification of Equations: In stability analysis, particularly when dealing with generators and transformers, you need to perform calculations that involve different voltage levels and power ratings.

By using the per-unit system, values like impedance, power, and voltage are normalized into a dimensionless form, making the equations simpler and easier to solve. You no longer have to worry about the specific units for different components of the system.

Uniformity in Systems: When multiple machines or components are involved, they often have different voltage and power ratings. The per-unit system converts these values to a common scale (usually based on the system's base values), making it easier to analyze the overall stability of the system.

Improving Comparisons: The per-unit system allows easier comparison of electrical quantities across different parts of the system, regardless of the absolute ratings of each component.

Fewer Units to Manage:Since the per-unit system is based on ratios, you typically only need to deal with dimensionless values (p.u. values), reducing the complexity and potential for unit-related errors during analysis.

Example:Consider a system with different components: One generator operates at 100 MVA, another at 200 MVA, and a third at 50 MVA. Using the per-unit system, you can convert all of these to a common base, making it easier to compare their electrical performance without worrying about their individual ratings.

VA rating (S) = 40MVA

voltage (V) = 11kV

Frequency (f) = 50Hz

Number of poles (P) = 4

Inertia constant (H) = 15 sec

Input power (Pin) = 20 MW

output power (Pout) = 15 MW

Acceleration power Pa = Pin Pout = 5 MW -

P_{a} = M * (d ^ 2 * delta)/(d * t ^ 2) = M*alpha

P_{a} = ((SH)/(180f)) * alpha

alpha = (5 * 10 ^ 6 * 180 * 50)/(40 * 10 ^ 6 * 15)

= 75°/s²

[1] Willaim D. Stevenson, Elements of Power System Analysis, Fouth Edition, McGraw-Hill
International Editions, Civil Engineering Series, McGraw-Hill Inc.

Inertia constant (H) is given by,

H= kinetic energy stored in rotor in MJ/ Machine rating in MVA(S) 

H = KE/S 

Calculation:

MVA rating (S) = 100 MVA

Given that, inertia constant (H) = 4 sec 

KE stored in rotor = HS = 4 * 100 = 400MJ. 

The wire required for DC system is lesser than AC distribution systems. So, we can save more copper through DC distribution system at higher voltage distribution. Hence direct current distribution system offers the best economy at high voltages. High voltage direct current (HVDC) power systems use D.C. for transmission of bulk power over long distances. For long-distance power transmission, HVDC lines are less expensive, and losses are less as compared to AC transmission. It interconnects the networks that have different frequencies and characteristics. HVDC transmission is economical only for long-distance transmission lines having a length of more than 600kms and for underground cables of length more than 50kms.

Economic Distance For HVDC distribution lines:

DC lines are cheaper than the AC lines, but the cost of DC terminal equipment is very high as compared to AC terminal cables as shown in the figure below. Thus, the initial cost is high in the HVDC transmission system, and it is low in the AC system.
The point where two curves meet is called the breakeven distance. Above the breakeven distance, the HVDC system becomes cheaper. Breakeven distance changes from 500 to 900 km in overhead transmission lines.
   

Advantages of HVDC distribution:

A lesser number of conductors and insulators are required thereby reducing the cost of the overall system.

It requires less phase to phase and ground to ground clearance.

Their towers are less costly and cheaper.

Corona loss is less as compared to HVAC transmission lines of similar power.

Power loss is reduced with DC because fewer numbers of lines are required for power transmission.

HVDC system uses earth return. If any fault occurs in one pole, the other pole with 'earth returns' behaves like an independent circuit. This results in a more flexible system.

HVDC acts as the asynchronous connection between two AC stations connected through an HVDC link, i.e. it interconnects two substations with different frequencies.

Due to the absence of frequency in the HVDC line, losses like skin effect and proximity effect does not occur in the system.

It does not generate or absorb any reactive power. So, there is no need for reactive power compensation.

Very accurate and lossless power flows through the DC link.

Disadvantages of HVDC distribution:

Converter substations are required at both the sending and the receiving end of the transmission lines, which result in increasing the cost.

Inverter and rectifier terminals generate harmonics which is reduced using active filters which are also very expensive.

The inverter used in Converter substations has limited overload capacity.

Circuit breakers are used in HVDC for circuit breaking, which is also very expensive.

It does not have transformers for changing the voltage levels.

The power transfer in long transmission lines is limited by the magnitude of the voltages at the two ends, the reactance between the two ends and the sine of the angle between the two voltages.

Maximum Steady-state power limit is given by P = EV/X

Important:

The power transfer capability of an AC transmission line is influenced by the stability limit; The power transfer capability of the existing transmission system can be enhanced through the improvement of stability limits.
We can improve steady-state stability limit by increasing transmission voltage (V) or by decreasing line reactance (X)

 Stored energy is calculated as K_E = H×S_rated

Here, H = 6.0 MJ/MVA, and S_rated = 200 MVA.
 So, K_E = 6.0×200=1200 MJ. 

The rotor acceleration is given by α_m=P_m−P_e/H

 Here, α_m=50/5=10 rad/s².

P _m,pu = 300/ 200 = 1.5; P _e,pu = 100/200 = 0.5;  ω,_pu (t) = ω(t)/ω_syn = ω _syn/ω_syn = 1.

dδ/dt = P_m−P_e/H×1/ω_syn

Substituting the given values:

dδ/dt = {(1.5−1.0)×6}/314 = 13.1 rad/s² = 13.1rad/s²

The synchronous speed ωsyn​ is calculated as:

ω_syn=2πf=2×3.14×60=377 rad/s

ω_m,syn= 2/P * ωsyn = 23.56 rad

The synchronous speed ωsyn​ is calculated as:

ω_syn = 2πf = 2×3.14×60 = 377 rad/s
The swing equation:
(2H/ω _sys) × ω_pu(t) × (d²δ(t)/dt²) = P_m,pu-P_e,pu

⇒(4/377) × ω_pu(t) × (d²δ(t)/dt²) = P_m,pu-P_e,pu

The insulation material in underground cables is critical to prevent electrical breakdown and to withstand mechanical, thermal, and chemical stresses. Historically, paper impregnated with oil (PILC) was used, but modern cables predominantly use XLPE due to its superior properties. XLPE provides excellent electrical insulation, is lightweight, and has a higher operating temperature limit (up to 90–110°C). It also resists moisture, which is crucial for underground and underwater applications where water ingress is a constant risk. PVC is used for low-voltage distribution but is unsuitable for high voltage due to lower thermal tolerance. Rubber insulation, while flexible, degrades over time and cannot withstand prolonged underground stress. XLPE insulation also reduces the need for thick sheathing and allows for more compact cable design. Additionally, XLPE exhibits excellent resistance to electrical treeing, mechanical pressure, and environmental conditions like soil chemicals, making it the standard for modern high-voltage underground and submarine cables. Proper insulation ensures long-term reliability, reduces maintenance, and allows higher current ratings. Underground cables are designed with multiple layers: conductor, insulation, metallic screen, and outer sheath. The choice of insulation directly influences the mechanical design because XLPE is semi-rigid, and the cable must accommodate bending, pulling, and thermal expansion during installation.

The metallic sheath in underground or underwater cables is typically made of copper or aluminum and serves multiple important functions. Its primary purpose is to provide a low-resistance path for fault currents, ensuring safety and enabling protective devices to operate efficiently. Fault currents occur when the cable insulation fails, allowing excess current to flow; the metallic sheath safely directs this current to the ground, preventing damage to the cable core. Another crucial function is moisture protection. Underground environments contain high levels of soil moisture, which can seep into the insulation, causing electrical degradation and reducing cable lifespan. The metallic sheath acts as a barrier to water ingress. It also provides a secondary electrical screen, reducing electromagnetic interference (EMI) for nearby circuits. While it contributes slightly to mechanical strength, it is not the primary purpose; mechanical protection is mainly provided by the armoring layer. For underwater cables, the metallic sheath is often lead-coated or corrugated aluminum, enhancing corrosion resistance while still allowing for bending during installation. Without a metallic sheath, underground cables would be prone to insulation failure, short circuits, and unsafe conditions. The sheath works in conjunction with other layers, including the insulation, bedding, and armoring, to ensure long-term durability and safety of power transmission.

Underground and submarine cables require conductors with high electrical conductivity, flexibility, and mechanical strength. The most common materials are copper and aluminum, each with advantages and limitations. Copper has higher conductivity (~5.8 × 10⁷ S/m), excellent ductility, and better fatigue resistance, making it ideal for compact, high-current, high-voltage applications. It also tolerates repeated bending during installation, which is essential for underground and submarine cabling where curves and turns are unavoidable. Aluminum, on the other hand, is lighter and cheaper than copper, which reduces installation costs and pulling forces. Aluminum is widely used for high-voltage underground and submarine cables where weight reduction is critical, such as long underwater spans or when the cable must be suspended on flexible supports. To maintain mechanical strength, aluminum conductors are often reinforced with steel wire cores. Both materials are compatible with modern insulation and sheathing techniques. Lead is sometimes used as an outer sheath material but never as a primary conductor due to low conductivity. In practice, choice between copper and aluminum depends on cost, mechanical requirements, voltage, and current ratings. Using the appropriate conductor ensures efficient energy transmission, minimal losses, and long-term operational reliability.

Underground and underwater cables are primarily used in situations where overhead lines are impractical due to space limitations, urban density, or environmental restrictions. In densely populated cities, installing overhead lines is often impossible due to the lack of physical space and the potential interference with buildings, roads, and other infrastructure. Similarly, in environmentally sensitive areas such as forests, water bodies, or scenic landscapes, overhead lines may not be allowed to prevent visual pollution and ecological disruption.

Underwater cables are commonly used to cross rivers, lakes, or seas, where installing pylons for overhead lines is technically challenging and expensive. For example, submarine power cables are used to connect islands or offshore wind farms to the mainland grid. These cables are designed to withstand high water pressure, mechanical stresses from currents, and potential hazards such as anchors or fishing activities.

While aesthetic reasons (option i) and safety can be considered additional benefits, they are not the primary drivers. Underground and underwater installations generally have higher initial costs than overhead lines, contrary to option iii, due to the need for excavation, insulation, and protective measures. Maintenance is also more challenging and costly compared to overhead lines (option iv), because accessing the cables requires excavation or specialized underwater operations.

The design of these cables involves careful consideration of electrical insulation, mechanical protection, thermal dissipation, and route planning. Materials used for insulation include cross-linked polyethylene (XLPE) or oil-impregnated paper, and mechanical protection may include steel armor for underwater applications. Ultimately, the use of underground or underwater cables is dictated by site-specific constraints, technical feasibility, and environmental regulations, making option (ii) the correct choice.

Submarine power cables are subjected to extreme mechanical stresses during installation and operation. These stresses include tensile forces during laying, abrasion against the seabed, impact from anchors, fishing trawls, or marine debris, and bending stresses due to ocean currents or uneven seabed topology. To withstand these hazards, submarine cables incorporate a metallic armor layer, typically made of steel wires or tapes, surrounding the insulated conductors.

The primary function of metallic armor is mechanical protection (option খ). This layer absorbs physical shocks and prevents the inner conductors and insulation from being damaged. Without the armor, a submarine cable could easily be punctured or crushed, leading to catastrophic electrical failure and significant repair costs. The armor must also provide tensile strength to withstand the forces applied during cable installation, which can involve pulling the cable over long distances at high tension.

It is important to note that the metallic armor does not provide electrical insulation (option ক). Insulation is already provided by the XLPE or paper layers surrounding the conductors. Similarly, while the metallic armor may slightly aid in thermal dissipation (option গ), its primary role is mechanical, not thermal. Option ঘ (reduce capacitance) is irrelevant because capacitance depends mainly on the conductor geometry and dielectric properties of the insulation, not the metallic armor.

The design of the armor also considers corrosion resistance, particularly for underwater cables. For example, galvanized steel is commonly used, and in some cases, copper or aluminum wires may be incorporated for additional electrical conductivity or grounding. By combining robust insulation with strong mechanical protection, submarine cables achieve a long service life, often exceeding 40 years, even in harsh marine environments.

While underground power cables offer advantages such as aesthetics, reduced exposure to weather, and lower risk of accidental contact, they also have notable limitations compared to overhead lines. The most significant limitation is higher initial cost (option ক).

The high cost of underground cables arises from several factors: **excavation of trenches

Installing long underground power cables in urban areas presents challenges such as existing infrastructure, roads, railways, and limited space. One effective method is Horizontal Directional Drilling (HDD), a trenchless technology that allows cables to be installed beneath obstacles without extensive excavation.

In HDD, a pilot hole is drilled along the planned route, then the hole is gradually enlarged, and the cable is pulled into the bore. This method minimizes traffic disruption, environmental impact, and surface restoration costs, making it ideal for densely built areas. HDD is widely used for high-voltage and medium-voltage distribution cables under rivers, highways, or railways.

Direct burial in open trenches (option i) is simpler and cheaper but is disruptive and often infeasible in dense urban settings. Aerial suspension (option iii) is used for overhead lines, not underground installations. Free-floating underwater placement (option iv) is specific to submarine cables, not urban underground projects.

HDD also allows for better cable protection since the cable is enclosed in conduits, reducing exposure to mechanical damage and moisture. Proper planning ensures bend radius, thermal expansion, and stress limits are maintained. HDD has become the preferred method for installing underground power infrastructure in modern urban settings.

Underground power cables have multiple layers, each designed for a specific purpose. After the conductor and insulation, cables typically include a metallic screen, bedding layer, and an outer sheath or jacket.

The outer sheath or jacket is usually made of polyethylene (PE), PVC, or other polymers, and its main purpose is to protect the cable mechanically from external damage caused by excavation, rocks, or soil movement. In addition, the sheath prevents moisture ingress, which could compromise the insulation and lead to partial discharge or breakdown.

While electrical insulation (option i) is provided by the conductor’s insulation (XLPE or paper), the sheath does not contribute significantly to dielectric properties. The sheath does not increase electrical conductivity (option iii) or reduce electromagnetic interference (option iv) — those are functions of conductor geometry, shielding, or metallic screens.

For submarine cables, the outer jacket may also provide resistance to corrosion, abrasion, and marine organisms, while armored cables include steel or copper armor under the jacket to further enhance mechanical protection. The design of the sheath must also consider thermal dissipation, flexibility, and compatibility with the installation environment.

In essence, the sheath and jacket layers serve as the first line of defense against physical and environmental damage, ensuring long-term reliability and safety of the cable system.

To find insulation thickness, we use:

R = ρ / (2πl) * ln(r2/r1)

Here, r1 = 1.25 cm, R = 495×10^6 Ω, l = 1000 m, ρ = 4.5×10^12 Ω-m.

Substituting:
495×10^6 = (4.5×10^12) / (2π×1000) * ln(r2/1.25)

Solving, ln(r2/1.25) ≈ 0.298 ⇒ r2 ≈ 2.5 cm.

Thus, insulation thickness = r2 − r1 = 2.5 − 1.25 = 1.25 cm.

This shows resistivity and insulation thickness are interdependent for achieving required insulation resistance. It also highlights the logarithmic relationship between radius ratio and resistance, crucial in high-voltage cable design.

The capacitance of a cable is given by the formula:

C = (εr *  * l) / (2π) * [14.4 / log₁₀(D/d)] * 10^-9

where:
C = Capacitance of the cable (F)
εr = Relative permittivity of the insulation
ε0 = Permittivity of free space = 8.854 × 10^-12 F/m
l = Length of the cable (m)
D = Overall diameter including insulation (cm)
d = Conductor diameter (cm)

Given data:
εr = 4
l = 1000 m
D = 1.8 cm
d = 1 cm

Substituting values into the formula:

C = (4 × 1000 ) / [41.4 × log₁₀(1.8 / 1)] × 10^-9
C ≈ 0.378 × 10^-6 F
C = 0.378 μF

The maximum stress occurs at the conductor surface and its value is given by:

g_max = (2V) / (d × log_e(D/d))   kV/cm

Where:
V = Operating voltage (r.m.s)
d = Conductor diameter (cm)
D = Inside diameter of sheath (cm)

Given:
V = 33 kV (r.m.s)
d = 1 cm
D = 4 cm

Substituting values:

g_max = (2 × 33) / (1 × log_e(4/1)) kV/cm
       ≈ 47.61 kV/cm (r.m.s)

The minimum stress occurs at the sheath and its value is given by:

g_min = (2V) / (D × log_e(D/d))   kV/cm

Substituting values:

g_min = (2 × 33) / (4 × log_e(4/1)) kV/cm
       ≈ 11.9 kV/cm (r.m.s)

Alternatively:

g_min = g_max × (d/D)
      = 47.61 × (1/4)
      = 11.9 kV/cm (r.m.s)