Electric Power Systems - Complete Guide

1. Introduction to Power Systems

1.1 Overview

An electric power system is an interconnected network for delivering electricity from producers to consumers. It consists of generation, transmission, distribution, and utilization components.

Main Components:

Generation: Power plants (thermal, hydro, nuclear, renewable)
Transmission: High voltage (110 kV to 765 kV) long-distance transfer
Distribution: Medium to low voltage (33 kV to 415 V) local delivery
Utilization: End-user consumption

1.2 Structure of Power System

Generation Level

11 kV to 25 kV

Transmission Level

110 kV, 220 kV, 400 kV, 765 kV

Distribution Level

33 kV, 11 kV, 415 V

2. Three-Phase Systems

2.1 Three-Phase Generation

Three-phase systems use three voltage sources with equal magnitude but 120° phase difference, providing constant power and efficient transmission.

Phase Voltages:

$ V_a = V_m \sin(\omega t) $
$ V_b = V_m \sin(\omega t - 120^\circ) $
$ V_c = V_m \sin(\omega t - 240^\circ) $

2.2 Star (Y) Connection

Line to Phase Relationships:

$ V_L = \sqrt{3} \times V_{ph} $
$ I_L = I_{ph} $

Where:
$ V_L $ = Line Voltage
$ V_{ph} $ = Phase Voltage
$ I_L $ = Line Current
$ I_{ph} $ = Phase Current

2.3 Delta (Δ) Connection

Line to Phase Relationships:

$ V_L = V_{ph} $
$ I_L = \sqrt{3} \times I_{ph} $

2.4 Three-Phase Power

Active Power (3-phase):

$ P = \sqrt{3} \times V_L \times I_L \times \cos(\phi) $
$ P = 3 \times V_{ph} \times I_{ph} \times \cos(\phi) $

Reactive Power (3-phase):

$ Q = \sqrt{3} \times V_L \times I_L \times \sin(\phi) $
$ Q = 3 \times V_{ph} \times I_{ph} \times \sin(\phi) $

Apparent Power (3-phase):

$ S = \sqrt{3} \times V_L \times I_L $
$ S = \sqrt{P^2 + Q^2} $

3. Transmission Line Parameters

3.1 Resistance (R)

DC Resistance:

$ R = \rho \times \dfrac{l}{A} $

Where:
$ \rho $ = Resistivity of conductor (Ω·m)
$ l $ = Length of conductor (m)
$ A $ = Cross-sectional area (m²)

AC Resistance (considering skin effect):

$ R_{ac} = R_{dc} \times (1 + y_s + y_p) $

Where:
$ y_s $ = Skin effect factor
$ y_p $ = Proximity effect factor

3.2 Inductance (L)

Inductance of Single-Phase Line:

$ L = 2 \times 10^{-7} \times \ln\left(\dfrac{D}{r}\right) \, \text{H/m} $

Where:
$ D $ = Distance between conductors (m)
$ r $ = Radius of conductor (m)

Inductance per phase (3-phase, symmetrical):

$ L = 2 \times 10^{-7} \times \ln\left(\dfrac{D_{eq}}{r'}\right) \, \text{H/m} $
$ D_{eq} = \sqrt[3]{D_{12} \times D_{23} \times D_{31}} $
$ r' = r \times e^{-1/4} $ (GMR - Geometric Mean Radius)

Inductive Reactance:

$ X_L = 2\pi fL = \omega L \, (\Omega) $

3.3 Capacitance (C)

Capacitance of Single-Phase Line:

$ C = \dfrac{\pi \epsilon_0 \epsilon_r}{\ln\left(\dfrac{D}{r}\right)} \, \text{F/m} $
$ C = \dfrac{2\pi \epsilon_0}{\ln\left(\dfrac{D}{r}\right)} \, \text{F/m} $ (for air, $ \epsilon_r = 1 $)

Where:
$ \epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} $ (permittivity of free space)

Capacitance per phase (3-phase):

$ C = \dfrac{2\pi \epsilon_0}{\ln\left(\dfrac{D_{eq}}{r}\right)} \, \text{F/m} $

Capacitive Reactance:

$ X_C = \dfrac{1}{2\pi fC} = \dfrac{1}{\omega C} \, (\Omega) $

Note: Capacitance effect is negligible for short lines (< 80 km) but becomes significant in medium (80-250 km) and long lines (> 250 km).

4. Performance of Transmission Lines

4.1 Classification of Transmission Lines

Type	Length	Model
Short Line	< 80 km	Resistance and Inductance only
Medium Line	80 - 250 km	Nominal π or T model
Long Line	> 250 km	Exact (distributed parameters)

4.2 Short Transmission Line

Capacitance is neglected. Only series resistance R and inductance L are considered.

Sending End Voltage and Current:

$ V_S = V_R + I_R Z $
$ I_S = I_R $

Where:
$ Z = R + jX_L $ (Series impedance)

ABCD Parameters (Short Line):

$ A = 1 $, $ B = Z $
$ C = 0 $, $ D = 1 $

$ V_S = A V_R + B I_R $
$ I_S = C V_R + D I_R $

4.3 Medium Transmission Line

4.3.1 Nominal π Model

ABCD Parameters (Nominal π):

$ A = D = 1 + \dfrac{YZ}{2} $
$ B = Z $
$ C = Y \left(1 + \dfrac{YZ}{4}\right) $

Where:
$ Y = j\omega C $ (Shunt admittance)
$ Z = R + jX_L $ (Series impedance)

4.3.2 Nominal T Model

ABCD Parameters (Nominal T):

$ A = D = 1 + \dfrac{YZ}{2} $
$ B = Z \left(1 + \dfrac{YZ}{4}\right) $
$ C = Y $

4.4 Long Transmission Line

Uses exact transmission line equations with distributed parameters.

Characteristic Impedance:

$ Z_c = \sqrt{\dfrac{Z}{Y}} = \sqrt{\dfrac{R + j\omega L}{j\omega C}} \, \Omega $

Propagation Constant:

$ \gamma = \sqrt{ZY} = \alpha + j\beta $

$ \alpha $ = Attenuation constant (Np/km)
$ \beta $ = Phase constant (rad/km)

ABCD Parameters (Long Line):

$ A = D = \cosh(\gamma l) $
$ B = Z_c \sinh(\gamma l) $
$ C = \dfrac{\sinh(\gamma l)}{Z_c} $

Where $ l $ = length of line

For Lossless Line (R = 0):

$ Z_c = \sqrt{\dfrac{L}{C}} $ (Real, resistive)
$ \gamma = j\beta = j\omega \sqrt{LC} $
$ A = D = \cos(\beta l) $
$ B = jZ_c \sin(\beta l) $
$ C = \dfrac{j \sin(\beta l)}{Z_c} $

5. Voltage Regulation and Efficiency

5.1 Voltage Regulation

Percentage Voltage Regulation:

$ \%VR = \left[\dfrac{V_{R(NL)} - V_{R(FL)}}{V_{R(FL)}}\right] \times 100 $
$ \%VR = \left[\dfrac{V_S / A - V_R}{V_R}\right] \times 100 $

Where:
$ V_{R(NL)} $ = Receiving end voltage at no load
$ V_{R(FL)} $ = Receiving end voltage at full load

Good voltage regulation: Lower percentage values indicate better regulation. Ideal value is 0%.

5.2 Transmission Efficiency

Efficiency:

$ \eta = \left(\dfrac{P_R}{P_S}\right) \times 100\% $
$ \eta = \left[\dfrac{P_R}{P_R + P_{loss}}\right] \times 100\% $

Where:
$ P_R $ = Power delivered (receiving end)
$ P_S $ = Power sent (sending end)
$ P_{loss} $ = $ I^2 R $ losses

Power Loss in Line:

$ P_{loss} = 3I^2R $ (for 3-phase line)
$ P_{loss} = 3 \left(\dfrac{P}{\sqrt{3} V_L \cos \phi}\right)^2 \times R $

6. Power Flow Through Transmission Lines

6.1 Power Equations

Real Power Transfer:

$ P = \dfrac{V_S V_R}{X} \sin(\delta) $

Where:
$ V_S $ = Sending end voltage
$ V_R $ = Receiving end voltage
$ X $ = Line reactance
$ \delta $ = Power angle (load angle)

Reactive Power:

$ Q_R = \dfrac{V_R}{X} (V_S \cos \delta - V_R) $
$ Q_S = \dfrac{V_S}{X} (V_S - V_R \cos \delta) $

6.2 Maximum Power Transfer

Maximum Power (at $ \delta = 90^\circ $):

$ P_{max} = \dfrac{V_S V_R}{X} $

Stability Consideration: Operating power angle $ \delta $ should be much less than $ 90^\circ $ for system stability (typically < 30-40°).

6.3 Circle Diagrams

Circle diagrams graphically represent the locus of receiving or sending end power for various operating conditions.

The sending end and receiving end power circles help visualize power transfer capability and operating limits.

7. Corona Effect

7.1 Corona Phenomenon

Corona is a luminous electrical discharge surrounding conductors when voltage gradient exceeds the breakdown strength of air (approximately 30 kV/cm at standard conditions).

Critical Disruptive Voltage (Peek's Formula):

$ V_c = 21.2 \, m_0 \, \delta \, r \ln\left(\dfrac{D}{r}\right) \, \text{kV (rms)} $

Where:
$ m_0 $ = Surface condition factor (0.8-1.0)
$ \delta $ = Air density factor $ = \dfrac{3.92b}{273 + t} $
$ b $ = Barometric pressure (cm of Hg)
$ t $ = Temperature (°C)
$ r $ = Radius of conductor (cm)
$ D $ = Distance between conductors (cm)

Visual Critical Voltage:

$ V_v = m_v V_c $
$ m_v = 1.0 $ for rough conductors
$ m_v = 0.72 $ for polished conductors

7.2 Corona Power Loss (Peek's Formula)

Corona Loss:

$ P_c = \dfrac{244}{\delta} (f + 25) \sqrt{\dfrac{r}{D}} (V - V_c)^2 \times 10^{-5} \, \text{kW/km/phase} $

Where:
$ f $ = Frequency (Hz)
$ V $ = Operating voltage (kV, rms)

Effects of Corona:

Power loss (undesirable)
Audible noise
Radio interference
Ozone production
Positive: Reduces voltage surges

7.3 Methods to Reduce Corona

Increase conductor diameter
Use bundled conductors
Increase spacing between conductors (limited by structural constraints)
Use hollow conductors

8. Sag and Tension in Overhead Lines

8.1 Sag Calculation

Sag is the vertical difference between the conductor at support point and lowest point of the catenary.

Sag (Supports at Same Level):

$ S = \dfrac{wL^2}{8T} $

Where:
$ w $ = Weight per unit length (kg/m or N/m)
$ L $ = Span length (m)
$ T $ = Tension in conductor (N)

Sag (Supports at Different Levels):

$ S_1 = \dfrac{wx_1^2}{2T} $
$ S_2 = \dfrac{wx_2^2}{2T} $
$ x_1 + x_2 = L $
$ S_1 - S_2 = h $ (difference in support levels)

8.2 Tension Calculation

Maximum Tension:

$ T_{max} = T + wS $
$ T_{max} \approx T + \dfrac{w^2L^2}{8T} $ (for small sag)

Effect of Wind and Ice:

$ w_{total} = \sqrt{(w_{conductor} + w_{ice})^2 + w_{wind}^2} $

$ w_{wind} = P \times d $ (N/m)
$ P $ = Wind pressure (N/m²)
$ d $ = Diameter of conductor with ice (m)

8.3 Effect of Temperature

Change in Length:

$ \Delta L = L \, \alpha \, \Delta T $

Where:
$ \alpha $ = Coefficient of linear expansion (/°C)
$ \Delta T $ = Change in temperature (°C)

Safety Factor: Conductors are designed with safety factor of 2-3 to ensure mechanical strength under maximum loading conditions.

9. Insulators

9.1 Types of Insulators

Pin Type

Used up to 33 kV. Single unit insulator mounted on pin.

Suspension Type

Used for high voltages (> 33 kV). Multiple discs in series.

Strain Type

Used at dead ends and sharp curves. Withstands mechanical tension.

Shackle Type

Used for low voltage distribution lines.

9.2 Voltage Distribution in Suspension Insulators

Voltage Across nth Disc:

$ V_n = V_1 K^{n-1} $

Where:
$ K = \dfrac{m}{1 + m} $ = Capacitance ratio
$ m = \dfrac{C_s}{C} $ (shunt to self capacitance ratio)
$ V_1 $ = Voltage across disc nearest to line

9.3 String Efficiency

String Efficiency:

$ \eta = \left( \dfrac{V}{n V_{\text{max}}} \right) \times 100\% $

Where:
$ V $ = Total voltage across string
$ n $ = Number of discs
$ V_{\text{max}} $ = Maximum voltage (across disc nearest to line)

Methods to Improve String Efficiency:

Use of longer cross-arms
Grading of insulators (different capacitances)
Use of guard rings/grading rings

10. Underground Cables

10.1 Cable Construction

Types of Cables (by voltage):

Low Tension (LT): Up to 1 kV
High Tension (HT): 1 kV to 11 kV
Super Tension (ST): 11 kV to 33 kV
Extra High Tension (EHT): 33 kV to 132 kV
Extra Super Voltage: Above 132 kV

10.2 Insulation Resistance

Insulation Resistance:

$ R = \dfrac{\rho}{2 \pi l} \ln \left( \dfrac{r_2}{r_1} \right) \, \Omega $

Where:
$ \rho $ = Resistivity of insulation ($\Omega \cdot m$)
$ l $ = Length of cable (m)
$ r_1 $ = Radius of conductor (m)
$ r_2 $ = Radius of cable with insulation (m)

10.3 Capacitance of Cables

Capacitance (Single Core Cable):

$ C = \dfrac{2 \pi \epsilon_0 \epsilon_r}{\ln \left( \dfrac{r_2}{r_1} \right)} \, \text{F/m} $

Where:
$ \epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} $
$ \epsilon_r $ = Relative permittivity of insulation

For Paper Insulated Cable ($ \epsilon_r \approx 3.5 $):

$ C = \dfrac{0.0242}{\log_{10} \left( \dfrac{r_2}{r_1} \right)} \, \mu\text{F/km} $

10.4 Dielectric Stress in Cables

Electric Stress at radius x:

$ E_x = \dfrac{V}{x \ln \left( \dfrac{r_2}{r_1} \right)} \, \text{V/m} $

Where $ V $ = Voltage between conductor and sheath

Maximum Stress (at conductor surface, $ x = r_1 $):

$ E_{\text{max}} = \dfrac{V}{r_1 \ln \left( \dfrac{r_2}{r_1} \right)} \, \text{V/m} $

Minimum Stress (at sheath, $ x = r_2 $):

$ E_{\text{min}} = \dfrac{V}{r_2 \ln \left( \dfrac{r_2}{r_1} \right)} \, \text{V/m} $

Most Economical Conductor Radius:

$ \dfrac{r_2}{r_1} = e = 2.718 $
(When ratio is $ e $, stress is most uniform)

10.5 Grading of Cables

Grading is the process of achieving uniform electric stress distribution.

Capacitance Grading

Using different layers of insulation with different permittivities.

Intersheath Grading

Using metallic intersheaths between conductor and outer sheath.

For n intersheaths:

$ r_1, r_2, r_3, \ldots, r_{n+1} $ should be in geometric progression
$ \dfrac{r_{i+1}}{r_i} = \text{constant} $

11. Fault Analysis and Short Circuit Calculations

11.1 Types of Faults

Fault Type	Occurrence (%)	Severity
Single Line to Ground (LG)	70-80%	Moderate
Line to Line (LL)	15-20%	Severe
Double Line to Ground (LLG)	10%	Very Severe
Three Phase (LLL or LLLG)	2-3%	Most Severe

11.2 Symmetrical Three-Phase Fault

Fault Current (Neglecting Resistance):

$ I_f = \dfrac{V}{X_d''} $ (sub-transient)
$ I_f = \dfrac{V}{X_d'} $ (transient)
$ I_f = \dfrac{V}{X_d} $ (steady-state)

Where:
$ V $ = Pre-fault voltage
$ X_d'' $ = Sub-transient reactance
$ X_d' $ = Transient reactance
$ X_d $ = Synchronous reactance

11.3 Per Unit System

Per Unit Quantity:

$ \text{Per Unit Value} = \dfrac{\text{Actual Value}}{\text{Base Value}} $

Base relationships:
$ S_{\text{base}} = \sqrt{3} \times V_{\text{base}} \times I_{\text{base}} $
$ Z_{\text{base}} = \dfrac{V_{\text{base}}}{I_{\text{base}}} = \dfrac{V_{\text{base}}^2}{S_{\text{base}}} $
$ I_{\text{base}} = \dfrac{S_{\text{base}}}{\sqrt{3} \times V_{\text{base}}} $

Changing Base Values:

$ Z_{\text{pu(new)}} = Z_{\text{pu(old)}} \times \left( \dfrac{V_{\text{base(old)}}}{V_{\text{base(new)}}} \right)^2 \times \left( \dfrac{S_{\text{base(new)}}}{S_{\text{base(old)}}} \right) $

11.4 Short Circuit MVA Method

Fault Level (MVA):

$ \text{MVA}_f = \dfrac{\sqrt{3} \times V_L \times I_f}{10^6} $

For per unit reactance:
$ \text{MVA}_f = \dfrac{\text{MVA}_{\text{base}}}{X_{\text{pu}}} $

11.5 Symmetrical Components

Any unbalanced three-phase system can be resolved into three balanced components:

Positive Sequence: Balanced three-phase, normal rotation (abc)
Negative Sequence: Balanced three-phase, reverse rotation (acb)
Zero Sequence: Three equal phasors, in phase

Transformation Equations:

$ V_a = V_{a0} + V_{a1} + V_{a2} $
$ V_b = V_{a0} + a^2 V_{a1} + a V_{a2} $
$ V_c = V_{a0} + a V_{a1} + a^2 V_{a2} $

Where:
$ a = 1 \angle 120^\circ = e^{j \dfrac{2\pi}{3}} $
$ a^2 = 1 \angle 240^\circ = e^{j \dfrac{4\pi}{3}} $
$ 1 + a + a^2 = 0 $

Inverse Transformation:

$ V_{a0} = \dfrac{1}{3} (V_a + V_b + V_c) $
$ V_{a1} = \dfrac{1}{3} (V_a + a V_b + a^2 V_c) $
$ V_{a2} = \dfrac{1}{3} (V_a + a^2 V_b + a V_c) $

11.6 Unsymmetrical Fault Analysis

Single Line to Ground Fault

Fault Current:

$ I_f = \dfrac{3V}{Z_1 + Z_2 + Z_0} $

Sequence currents:
$ I_{a0} = I_{a1} = I_{a2} = \dfrac{V}{Z_1 + Z_2 + Z_0} $

Line to Line Fault

Fault Current:

$ I_f = \dfrac{\sqrt{3} V}{Z_1 + Z_2} $

Sequence currents:
$ I_{a0} = 0 $
$ I_{a1} = -I_{a2} = \dfrac{V}{Z_1 + Z_2} $

Double Line to Ground Fault

Fault Current:

$ I_f = I_b + I_c $
$ I_{a1} = \dfrac{V}{Z_1 + \dfrac{Z_2 Z_0}{Z_2 + Z_0}} $
$ I_{a2} = -I_{a1} \times \dfrac{Z_0}{Z_2 + Z_0} $
$ I_{a0} = -I_{a1} \times \dfrac{Z_2}{Z_2 + Z_0} $

12. Power Factor and Its Improvement

12.1 Power Factor

Power Factor:

$ \cos \phi = \dfrac{P}{S} = \dfrac{P}{\sqrt{P^2 + Q^2}} $
$ \tan \phi = \dfrac{Q}{P} $

Where:
$ P $ = Active Power (W, kW, MW)
$ Q $ = Reactive Power (VAR, kVAR, MVAR)
$ S $ = Apparent Power (VA, kVA, MVA)

12.2 Causes of Low Power Factor

Induction motors operating at no-load or light load
Transformers operating at light loads
Arc lamps, electric discharge lamps
Industrial heating furnaces
Welding equipment

12.3 Effects of Low Power Factor

Increased current for same power → Higher $ I^2R $ losses
Larger conductor size required
Greater voltage drop
Reduced system capacity
Higher electricity bills (penalty charges)
Poor voltage regulation

12.4 Power Factor Improvement

Capacitor Required for PF Improvement

Leading kVAR Required:

$ Q_C = P(\tan \phi_1 - \tan \phi_2) $ kVAR

Or:
$ Q_C = P \left[ \left( \dfrac{1}{\cos \phi_1} \right) - \left( \dfrac{1}{\cos \phi_2} \right) \right] \sin \phi_1 $

Where:
$ \phi_1 $ = Original power factor angle
$ \phi_2 $ = Desired power factor angle
$ P $ = Active power (kW)

Capacitance Value:

$ C = \dfrac{Q_C \times 10^3}{2 \pi f V^2} $ F
$ C = \dfrac{Q_C \times 10^9}{2 \pi f V^2} $ μF

For 3-phase:
$ C = \dfrac{Q_C \times 10^3}{3 \times 2 \pi f V_{ph}^2} $ F per phase

12.5 Methods of Power Factor Improvement

Static Capacitors

Most common method. Can be connected in shunt across load.

Synchronous Condensers

Over-excited synchronous motors. Provides variable reactive power.

Phase Advancers

AC exciter for slip-ring induction motors.

12.6 Advantages of Power Factor Improvement

Reduction in current and $ I^2R $ losses
Reduction in voltage drop
Increased system capacity
Reduced electricity bills
Improved voltage regulation
Reduced kVA demand

13. Switchgear and Protection

13.1 Circuit Breakers

A circuit breaker is an automatic device that can make, carry, and break currents under normal and abnormal (fault) conditions.

Circuit Breaker Ratings

Making Current:

$ I_{\text{make}} = 2.55 \times \text{Rated Breaking Current} $
(For asymmetrical fault current with DC component)

Breaking Capacity:

$ \text{MVA}_{\text{breaking}} = \dfrac{\sqrt{3} \times V_L \times I_{\text{break}}}{10^6} $

13.2 Arc Interruption

When contacts separate, an arc is drawn. Arc must be extinguished for successful interruption.

Methods of Arc Extinction:

High Resistance Method: Arc resistance is increased
Low Resistance/Current Zero Method: Arc is extinguished at current zero

13.3 Types of Circuit Breakers

Type	Medium	Voltage Range
Air Break CB	Air at atmospheric pressure	Up to 15 kV
Air Blast CB	Compressed air (20-30 bar)	132 kV to 400 kV
Oil CB (Bulk/Minimum)	Insulating oil	Up to 220 kV
SF₆ CB	Sulphur Hexafluoride gas	Up to 765 kV
Vacuum CB	Vacuum (10⁻⁴ to 10⁻⁷ torr)	11 kV to 33 kV

13.4 Protective Relays

Relays detect abnormal conditions and initiate circuit breaker tripping.

Types of Relays

Overcurrent Relay

Operates when current exceeds preset value. IDMT (Inverse Definite Minimum Time).

Differential Relay

Compares currents entering and leaving. Used for transformers, generators.

Distance Relay

Measures impedance. Used for transmission line protection.

Directional Relay

Operates only for fault current in specific direction.

13.5 Fuses

Fusing Factor:

$ \text{Fusing Factor} = \dfrac{\text{Minimum Fusing Current}}{\text{Current Rating}} $
(Typically 1.3 to 2.0)

Prospective Current:

$ I_p = \dfrac{V}{Z} $
(Current that would flow if fuse replaced by solid link)

Cut-off Current:

$ I_c < I_p $
(Actual maximum current reached before fuse melts)

14. Grounding (Earthing) Systems

14.1 Purpose of Grounding

Safety of personnel and equipment
Provide path for fault current to ground
Maintain voltage stability
Enable ground fault protection
Discharge lightning surges

14.2 Neutral Grounding Methods

Method	Application	Ground Fault Current
Solid Grounding	LV distribution, generators	Very high
Resistance Grounding	Medium voltage systems	Limited (200-400 A)
Reactance Grounding	Large generators	Limited
Peterson Coil (Resonant)	Distribution systems	Very low
Ungrounded	Rarely used	Capacitive current only

14.3 Earth Resistance

Resistance of Rod Electrode:

$ R = \dfrac{\rho}{2 \pi L} \left[ \ln \left( \dfrac{4L}{d} \right) - 1 \right] \, \Omega $

Where:
$ \rho $ = Soil resistivity ($\Omega \cdot m$)
$ L $ = Length of rod (m)
$ d $ = Diameter of rod (m)

Resistance of Plate Electrode:

$ R = \dfrac{\rho}{4 \sqrt{A}} \left( 1 + \dfrac{1}{1 + \dfrac{h}{\sqrt{A}}} \right) \, \Omega $

Where:
$ A $ = Area of one side of plate (m²)
$ h $ = Depth of burial (m)

Parallel Electrodes:

For $ n $ identical electrodes:
$ R_{\text{eq}} = \dfrac{R}{n \times \eta} $

Where $ \eta $ = efficiency factor (< 1 due to mutual interaction)

14.4 Step and Touch Potentials

Step Potential:

$ V_{\text{step}} = I_f \times R_{\text{step}} $
(Voltage between two feet 1 m apart)

Touch Potential:

$ V_{\text{touch}} = I_f \times R_{\text{touch}} $
(Voltage between hand and feet)

Safety Limits (for 0.1 sec exposure):
Step Potential: 2500 V
Touch Potential: 1000 V

15. Power System Stability

15.1 Types of Stability

Steady-State Stability: Ability to maintain synchronism under slow/gradual changes
Transient Stability: Ability to maintain synchronism after large disturbances (faults, switching)
Dynamic Stability: Ability to maintain synchronism under continuous small disturbances

15.2 Swing Equation

Swing Equation (Mechanical form):

$ M \dfrac{d^2 \delta}{dt^2} = P_m - P_e $
$ J \dfrac{d^2 \theta}{dt^2} = T_m - T_e $

Where:
$ M $ = Inertia constant (MJ-s/rad or MW-s/rad)
$ \delta $ = Rotor angle (electrical radians)
$ \theta $ = Rotor angle (mechanical radians)
$ P_m $ = Mechanical power input
$ P_e $ = Electrical power output
$ J $ = Moment of inertia (kg·m²)

In terms of H (Inertia Constant):

$ \dfrac{2H}{\omega_s} \dfrac{d^2 \delta}{dt^2} = P_m - P_e $

$ H = \dfrac{\text{Stored kinetic energy at rated speed}}{\text{Machine rating}} $
$ H = \dfrac{\dfrac{1}{2} J \omega_m^2}{S_{\text{rated}}} $ (MJ/MVA or seconds)

$ \omega_s $ = Synchronous speed (rad/s) = $ 2 \pi f $

15.3 Equal Area Criterion

For transient stability analysis of single machine infinite bus system.

Stability Condition:

$ A_1 = A_2 $

Where:
$ A_1 $ = Accelerating area (during fault)
$ A_2 $ = Decelerating area (after fault clearing)

Critical Clearing Angle:

$ \delta_{\text{cr}} = \cos^{-1} \left[ \dfrac{P_m}{P_{\text{max2}}} + \left( \pi - \delta_0 - \sin^{-1} \left( \dfrac{P_m}{P_{\text{max1}}} \right) \right) \right] $

Where:
$ \delta_0 $ = Initial operating angle
$ P_{\text{max1}} $ = Max power during fault
$ P_{\text{max2}} $ = Max power after fault cleared

15.4 Methods to Improve Stability

Fast fault clearing (high-speed circuit breakers)
Reduction in transfer reactance (series capacitors, parallel circuits)
High-speed excitation systems (AVR)
Single pole switching/reclosing
Dynamic braking (braking resistors)
Power system stabilizers (PSS)
FACTS devices (SVC, STATCOM, TCSC)

16. Load Flow (Power Flow) Analysis

16.1 Bus Classification

Bus Type	Known Quantities	Unknown Quantities
Slack/Swing Bus	$ \|V\|, \delta $ (reference)	$ P, Q $
PV Bus (Generator)	$ P, \|V\| $	$ Q, \delta $
PQ Bus (Load)	$ P, Q $	$ \|V\|, \delta $

16.2 Bus Admittance Matrix ($ Y_{\text{bus}} $)

$ Y_{\text{bus}} $ Formation:

$ Y_{ii} = \sum $ (admittances connected to bus $ i $)
$ Y_{ij} = - $ (admittance between bus $ i $ and $ j $)

For line with impedance $ Z_{ij} $ and shunt admittance $ Y_{\text{sh}} $:
$ Y_{ii} = Y_{ii} + \dfrac{1}{Z_{ij}} + \dfrac{Y_{\text{sh}}}{2} $
$ Y_{jj} = Y_{jj} + \dfrac{1}{Z_{ij}} + \dfrac{Y_{\text{sh}}}{2} $
$ Y_{ij} = Y_{ji} = -\dfrac{1}{Z_{ij}} $

16.3 Power Flow Equations

Complex Power at Bus $ i $:

$ S_i = P_i + jQ_i = V_i I_i^* $
$ S_i = V_i \left[ \sum (Y_{ij} V_j) \right]^* $

In polar form:
$ P_i = \sum |V_i||V_j||Y_{ij}| \cos(\theta_{ij} - \delta_i + \delta_j) $
$ Q_i = \sum |V_i||V_j||Y_{ij}| \sin(\theta_{ij} - \delta_i + \delta_j) $

16.4 Gauss-Seidel Method

Iterative Formula:

$ V_i^{(k+1)} = \dfrac{1}{Y_{ii}} \left[ \dfrac{P_i - jQ_i}{(V_i^{(k)})^*} - \sum (Y_{ij} V_j) \right] $

For $ j < i $: use $ V_j^{(k+1)} $ (latest values)
For $ j > i $: use $ V_j^{(k)} $ (previous values)

Acceleration Factor: Convergence can be improved using acceleration factor $ \alpha $ (typically 1.3-1.6):
$ V_i^{(k+1)} = V_i^{(k)} + \alpha (V_i^{\text{calculated}} - V_i^{(k)}) $

16.5 Newton-Raphson Method

Jacobian Matrix Form:

$ \begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} J_1 & J_2 \\ J_3 & J_4 \end{bmatrix} \begin{bmatrix} \Delta \delta \\ \Delta V \end{bmatrix} $

Where:
$ \Delta P_i = P_{i,\text{spec}} - P_{i,\text{calc}} $
$ \Delta Q_i = Q_{i,\text{spec}} - Q_{i,\text{calc}} $
$ J_1 = \dfrac{\partial P}{\partial \delta}, \quad J_2 = \dfrac{\partial P}{\partial V} $
$ J_3 = \dfrac{\partial Q}{\partial \delta}, \quad J_4 = \dfrac{\partial Q}{\partial V} $

Convergence: Newton-Raphson converges faster (quadratic convergence) compared to Gauss-Seidel (linear convergence) but requires more computation per iteration.

16.6 Fast Decoupled Load Flow

Simplified version of Newton-Raphson using assumptions:

$ P $ is mainly dependent on $ \delta $ (angle)
$ Q $ is mainly dependent on $ |V| $ (magnitude)

Decoupled Equations:

$ \Delta P = [B'] \Delta \delta $
$ \Delta Q = [B''] \Delta V $

Where $ B' $ and $ B'' $ are constant matrices

17. Economic Operation of Power Systems

17.1 Input-Output Characteristics

Heat Rate:

$ \text{Heat Rate} = \dfrac{\text{Heat Input}}{\text{Electrical Output}} $
$ HR = \dfrac{H}{P} $ (kJ/kWh or Btu/kWh)

Fuel Cost:

$ \text{Cost} = C_i(P_i) = a_i + b_i P_i + c_i P_i^2 $ ($/hr)

Where:
$ P_i $ = Power output of generator $ i $
$ a_i, b_i, c_i $ = Cost coefficients

17.2 Incremental Cost

Incremental Fuel Cost:

$ IC_i = \dfrac{dC_i}{dP_i} = b_i + 2c_i P_i $ ($/MWh)

17.3 Economic Load Dispatch (Without Loss)

Objective: Minimize total fuel cost while satisfying load demand.

Optimization Problem:

Minimize: $ C_T = \sum C_i(P_i) $
Subject to: $ \sum P_i = P_D $ (load demand)
$ P_{i,\text{min}} \leq P_i \leq P_{i,\text{max}} $

Condition for Economic Dispatch:

$ IC_1 = IC_2 = IC_3 = \dots = IC_n = \lambda $

Where $ \lambda $ = Lagrange multiplier ($/MWh)
(All incremental costs must be equal)

Solution:

$ \dfrac{dC_i}{dP_i} = \lambda $
$ b_i + 2c_i P_i = \lambda $
$ P_i = \dfrac{\lambda - b_i}{2c_i} $

Find $ \lambda $ such that: $ \sum P_i = P_D $

17.4 Economic Dispatch with Transmission Loss

Coordination Equations:

$ \dfrac{dC_i/dP_i}{1 - \partial P_L/\partial P_i} = \lambda $

Or: $ IC_i \times L_i = \lambda $

Where:
$ L_i = \dfrac{1}{1 - \partial P_L/\partial P_i} $ = Penalty factor

Transmission Loss (B-Coefficients):

$ P_L = \sum \sum P_i B_{ij} P_j + \sum B_{0i} P_i + B_{00} $

$ \dfrac{\partial P_L}{\partial P_i} = 2 \sum B_{ij} P_j + B_{0i} $

17.5 Unit Commitment

Determining which generating units to operate and when to start/shut down units to minimize cost over a period.

Constraints:

Load demand must be met
Spinning reserve requirements
Minimum up/down time constraints
Startup and shutdown costs
Ramp rate limits

Methods: Priority list method, Dynamic programming, Lagrangian relaxation, Genetic algorithms

18. Voltage Control in Power Systems

18.1 Need for Voltage Control

Maintain voltage within acceptable limits (±5% typically)
Improve system stability
Reduce losses
Proper operation of equipment

18.2 Methods of Voltage Control

18.2.1 Excitation Control

Automatic Voltage Regulators (AVR) control generator field current to maintain terminal voltage.

Reactive Power from Generator:

$ Q_G = \dfrac{EV}{X_s} \sin(\delta) - \dfrac{V^2}{X_s} $

Where:
$ E $ = Internal voltage (excitation dependent)
$ V $ = Terminal voltage
$ X_s $ = Synchronous reactance
$ \delta $ = Power angle

18.2.2 Tap-Changing Transformers

Off-Load Tap Changer (OLTC):

$ V_2 = V_1 \times \dfrac{N_2}{N_1} = \dfrac{V_1}{a} $

Where $ a $ = turns ratio

On-Load Tap Changer:

Tap range: typically ±10% to ±20%
Step size: 1.25% to 2.5%
Number of taps: 17, 33, etc.

18.2.3 Shunt Compensation

Shunt Capacitor:

$ Q_C = \dfrac{V^2}{X_C} = V^2 \omega C $ (leading VAR)
(Raises voltage by supplying reactive power)

Shunt Reactor:

$ Q_L = \dfrac{V^2}{X_L} = \dfrac{V^2}{\omega L} $ (lagging VAR)
(Lowers voltage by absorbing reactive power)

18.2.4 Series Compensation

Series Capacitor:

$ X_{\text{eff}} = X_L - X_C $
Degree of compensation = $ \dfrac{X_C}{X_L} \times 100\% $

Benefits:
- Reduces effective line reactance
- Improves voltage regulation
- Increases power transfer capability

18.3 Static VAR Compensators (SVC)

FACTS devices for dynamic reactive power compensation.

TCR (Thyristor Controlled Reactor)

Variable inductive reactance by controlling firing angle

TSC (Thyristor Switched Capacitor)

Switched capacitor banks

STATCOM

Voltage source converter based, faster response than SVC

19. Frequency and Active Power Control

19.1 Load Frequency Control

Maintains system frequency at nominal value (50 Hz or 60 Hz) by balancing generation and load.

Frequency-Power Relationship:

$ \Delta f \propto (P_G - P_D - P_L) $

Where:
$ P_G $ = Total generation
$ P_D $ = Load demand
$ P_L $ = Losses

19.2 Governor Characteristics

Speed Regulation (Droop):

$ R = \dfrac{f_{\text{NL}} - f_{\text{FL}}}{f_{\text{rated}} \times P_{\text{rated}}} $
$ R = \dfrac{\Delta f}{f_0 \times \Delta P_G} $ (per unit)

Typical value: $ R = 0.04 $ to $ 0.06 $ (4-6%)

Where:
$ f_{\text{NL}} $ = No-load frequency
$ f_{\text{FL}} $ = Full-load frequency

Governor Response:

$ \Delta P_G = -\dfrac{1}{R} \times \dfrac{\Delta f}{f_0} $

19.3 Load Damping

Load-Frequency Characteristic:

$ \Delta P_D = D \times \Delta f $

Where $ D $ = Load damping constant (typically 1-2% per Hz)

19.4 Control Area Concept

Area Control Error (ACE):

$ \text{ACE} = \Delta P_{\text{tie}} + B \times \Delta f $

Where:
$ \Delta P_{\text{tie}} $ = Change in tie-line power
$ B $ = Frequency bias factor

19.5 Control Strategies

Primary Control: Governor action (automatic, fast, 0-30 sec)
Secondary Control: AGC (Automatic Generation Control) (30 sec - few minutes)
Tertiary Control: Economic dispatch (minutes to hours)

20. HVDC Transmission

20.3 HVDC Converter Types

Line Commutated Converter (LCC)

Uses thyristors (SCR)
Requires AC voltage for commutation
Consumes reactive power
Mature technology, lower cost

Voltage Source Converter (VSC)

Uses IGBTs or GTOs
Self-commutated
Can control both active and reactive power independently
Can connect to weak AC systems
Higher losses, higher cost

20.4 HVDC Power Equations

DC Power Transfer:

$ P_{\text{dc}} = V_d \times I_d $

Where:
$ V_d $ = DC voltage
$ I_d $ = DC current

DC Voltage (6-pulse bridge):

$ V_{d0} = \dfrac{3\sqrt{2}}{\pi} \times V_{\text{LL}} = 1.35 \times V_{\text{LL}} $
$ V_d = V_{d0} \cos(\alpha) - R_c I_d $

Where:
$ \alpha $ = Firing angle (delay angle)
$ R_c $ = Commutation resistance
$ V_{\text{LL}} $ = AC line-to-line voltage

For 12-pulse bridge:

$ V_{d0} = \dfrac{3\sqrt{6}}{\pi} \times V_{\text{LL}} = 2.70 \times V_{\text{LL}} $

20.5 HVDC Control

Control Modes:

Rectifier: Constant current control ($ \alpha = 5^\circ $ to $ 15^\circ $)
Inverter: Constant extinction angle control ($ \gamma = 15^\circ $ to $ 20^\circ $) or constant voltage control

21. FACTS (Flexible AC Transmission Systems)

21.1 Introduction

FACTS devices are power electronic-based systems that enhance controllability and increase power transfer capability of AC transmission networks.

21.2 Classification of FACTS Devices

Series Controllers

TCSC

Thyristor Controlled Series Capacitor - Variable series compensation

SSSC

Static Synchronous Series Compensator - Voltage source based

Shunt Controllers

SVC

Static VAR Compensator - TCR + TSC/FC

STATCOM

Static Synchronous Compensator - Voltage source based, faster response

Combined Controllers

UPFC

Unified Power Flow Controller - Most versatile FACTS device

IPFC

Interline Power Flow Controller - Controls power flow in multiple lines

21.3 Benefits of FACTS

Increased transmission capacity
Improved voltage regulation
Enhanced transient and steady-state stability
Damping of power oscillations
Reduced loop flows
Improved power quality

21.4 SVC Characteristics

TCR Susceptance:

$ B_{\text{TCR}} = \dfrac{2(\pi - \alpha) + \sin(2\alpha)}{\pi X_L} $

Where $ \alpha $ = Firing angle ($ 90^\circ \leq \alpha \leq 180^\circ $)
$ \alpha = 90^\circ $: Full conduction (max susceptance)
$ \alpha = 180^\circ $: No conduction (zero susceptance)

SVC Reactive Power:

$ Q_{\text{SVC}} = V^2 \times B_{\text{SVC}} $
$ B_{\text{SVC}} = B_{\text{TCR}} + B_C $

(Capacitive $ Q $ is positive, Inductive $ Q $ is negative)

21.5 STATCOM

STATCOM Current:

$ I_{\text{STATCOM}} = \dfrac{V_s - V_t}{X_L} $

Where:
$ V_s $ = STATCOM output voltage (controllable)
$ V_t $ = Terminal voltage
$ X_L $ = Coupling reactance

Reactive Power:

$ Q = V_t \dfrac{V_s - V_t}{X_L} $

If $ V_s > V_t $: Capacitive (leading $ Q $)
If $ V_s < V_t $: Inductive (lagging $ Q $)

21.6 TCSC

Effective Reactance:

$ X_{\text{eff}} = X_L - X_{\text{TCSC}} $

$ X_{\text{TCSC}} $ is controllable by firing angle

Power Flow with TCSC:

$ P = \dfrac{V_S V_R}{X_L - X_{\text{TCSC}}} \sin(\delta) $

Increased power transfer capability

22. Renewable Energy Integration

22.1 Challenges of Renewable Integration

Intermittency: Variable generation from wind and solar
Low Inertia: Inverter-based resources have no rotating mass
Voltage Control: Fluctuating output affects voltage stability
Harmonics: Power electronic converters introduce harmonics
Forecasting: Accurate prediction of generation is difficult
Grid Codes: Need to meet strict interconnection requirements

22.2 Wind Power Integration

Wind Turbine Power

Available Wind Power:

$ P_{\text{wind}} = \dfrac{1}{2} \times \rho \times A \times v^3 $

Where:
$ \rho $ = Air density (≈ 1.225 kg/m³ at sea level)
$ A $ = Swept area of blades = $ \pi R^2 $ (m²)
$ v $ = Wind speed (m/s)

Turbine Power Output:

$ P_{\text{turbine}} = \dfrac{1}{2} \times \rho \times A \times C_p \times v^3 $

Where:
$ C_p $ = Power coefficient (max ≈ 0.59, Betz limit)
Typical $ C_p $ = 0.35 to 0.45

Tip Speed Ratio:

$ \lambda = \dfrac{\omega \times R}{v} $

Where:
$ \omega $ = Angular velocity of blade (rad/s)
$ R $ = Blade radius (m)

Types of Wind Generators

Type	Description	Speed Control
Type 1	Fixed-speed SCIG	No speed control
Type 2	SCIG with variable rotor resistance	Limited speed control
Type 3	DFIG (Doubly-Fed Induction Generator)	±30% speed variation
Type 4	Full converter (PMSG or SCIG/WRSG)	Full speed variation

22.3 Solar PV Integration

Solar Irradiance Power:

$ P_{\text{available}} = G \times A $

Where:
$ G $ = Solar irradiance (W/m²)
$ A $ = Panel area (m²)

PV Panel Output:

$ P_{\text{PV}} = G \times A \times \eta \times \text{PR} $

Where:
$ \eta $ = Panel efficiency (15-22% typically)
$ \text{PR} $ = Performance ratio (0.75-0.85)

Temperature Effect:

$ P_T = P_{\text{STC}} \times [1 - \beta (T_{\text{cell}} - 25)] $

Where:
$ \beta $ = Temperature coefficient (≈ -0.4 to -0.5%/°C)
$ T_{\text{cell}} $ = Cell temperature (°C)
$ P_{\text{STC}} $ = Power at standard test conditions

22.4 Energy Storage Systems

Critical for managing renewable intermittency and providing grid services.

Battery Storage

Li-ion, Lead-acid, Flow batteries - Fast response, scalable

Pumped Hydro

Largest capacity, 70-85% efficiency, site-specific

Compressed Air

CAES - Large scale storage, requires geological formations

Flywheel

High power, fast response, short duration

Energy Storage Capacity:

$ E = P \times t \times \eta $

Where:
$ E $ = Energy capacity (kWh or MWh)
$ P $ = Power rating (kW or MW)
$ t $ = Discharge duration (hours)
$ \eta $ = Round-trip efficiency

23. Smart Grid Technologies

23.1 Smart Grid Concepts

Modern electrical grid using digital communication and control technologies for improved efficiency, reliability, and sustainability.

23.2 Key Components

Advanced Metering Infrastructure (AMI): Smart meters with two-way communication
Distribution Automation: Automated switches, reclosers, and fault detection
Demand Response: Dynamic pricing and load management
Wide Area Monitoring: PMUs (Phasor Measurement Units) for real-time monitoring
Distributed Generation: Integration of DERs (Distributed Energy Resources)
Energy Management Systems: SCADA, EMS, DMS integration

23.3 Microgrids

Localized grids that can operate independently or in conjunction with the main grid.

Microgrid Modes:

Grid-Connected Mode: Operates parallel with utility grid
Islanded Mode: Operates autonomously during grid outages

23.4 Virtual Power Plants (VPP)

Cloud-based aggregation of distributed energy resources to act as a single power plant.

23.5 Benefits of Smart Grid

Improved reliability and resilience
Better integration of renewable energy
Enhanced energy efficiency
Reduced peak demand through demand response
Faster outage detection and restoration
Consumer empowerment and participation
Reduced carbon emissions

📚 Quick Reference Summary

Key Formulas - Three Phase Systems

Star: $ V_L = \sqrt{3} V_{\text{ph}} $, $ I_L = I_{\text{ph}} $
Delta: $ V_L = V_{\text{ph}} $, $ I_L = \sqrt{3} I_{\text{ph}} $
3φ Power: $ P = \sqrt{3} V_L I_L \cos \phi $

Key Formulas - Transmission Lines

Inductance: $ L = 2 \times 10^{-7} \ln(D_{\text{eq}}/r') $ H/m
Capacitance: $ C = \dfrac{2 \pi \epsilon_0}{\ln(D_{\text{eq}}/r)} $ F/m
Characteristic Impedance: $ Z_c = \sqrt{Z/Y} $
Voltage Regulation: $ \% \text{VR} = \left( \dfrac{V_{\text{NL}} - V_{\text{FL}}}{V_{\text{FL}}} \right) \times 100 $

Key Formulas - Power Flow

Power Transfer: $ P = \dfrac{V_S V_R}{X} \sin \delta $
Maximum Power: $ P_{\text{max}} = \dfrac{V_S V_R}{X} $ (at $ \delta = 90^\circ $)
Corona Loss: $ P_c = \dfrac{244}{\delta} (f+25) \sqrt{\dfrac{r}{D}} (V - V_c)^2 \times 10^{-5} $ kW/km

Key Formulas - Fault Analysis

Per Unit: $ Z_{\text{pu}} = \dfrac{Z_{\text{actual}}}{Z_{\text{base}}} $
$ Z_{\text{base}} = \dfrac{V_{\text{base}}^2}{S_{\text{base}}} $
Symmetrical Fault: $ I_f = \dfrac{V}{Z} $ (3-phase)
LG Fault: $ I_f = \dfrac{3V}{Z_1 + Z_2 + Z_0} $

Key Formulas - Power Factor

PF = $ \cos \phi = \dfrac{P}{S} = \dfrac{P}{\sqrt{P^2 + Q^2}} $
Capacitor kVAR: $ Q_C = P (\tan \phi_1 - \tan \phi_2) $
Capacitance: $ C = \dfrac{Q_C \times 10^9}{2 \pi f V^2} $ μF

Key Formulas - Economic Dispatch

Cost Function: $ C(P) = a + bP + cP^2 $
Incremental Cost: $ IC = \dfrac{dC}{dP} = b + 2cP $
Optimal Condition: $ IC_1 = IC_2 = \dots = \lambda $

Important Constants

Permittivity: $ \epsilon_0 = 8.854 \times 10^{-12} $ F/m
Permeability: $ \mu_0 = 4 \pi \times 10^{-7} $ H/m
Standard Frequency: 50 Hz (most countries), 60 Hz (US, parts of Americas)
Air Breakdown: ≈ 30 kV/cm (at standard conditions)

🎯 Important Topics for Exams

High Priority Topics:

Three-phase system calculations (Star/Delta conversions, power)
Transmission line parameters (R, L, C calculations)
Performance of short, medium, and long transmission lines (ABCD parameters)
Voltage regulation and efficiency calculations
Symmetrical and unsymmetrical fault analysis
Per unit system and base conversions
Power factor improvement calculations
Economic load dispatch (with and without losses)
Sag and tension calculations
Corona effect and power loss
Insulator string efficiency
Load flow analysis methods (Gauss-Seidel, Newton-Raphson)
Power system stability (swing equation, equal area criterion)
Grounding/earthing systems
HVDC vs HVAC comparison

Problem-Solving Tips:

Always check units and convert to consistent system
Draw circuit diagrams for complex problems
Use per unit system for multi-voltage level problems
Remember phasor notation and angle conventions
Check your answers for reasonableness (efficiency < 100%, PF ≤ 1, etc.)
Practice numerical problems from each topic
Understand physical concepts, not just formulas

Bus Type	Known Quantities	Unknown Quantities
Slack/Swing Bus	\( \|V\|, \delta \) (reference)	\( P, Q \)
PV Bus (Generator)	\( P, \|V\| \)	\( Q, \delta \)
PQ Bus (Load)	\( P, Q \)	\( \|V\|, \delta \)