Module 8 of 12

Power Electronics & Renewable Energy Systems

Advanced Power Conversion, Energy Storage, and Sustainable Power Generation

1. Power Semiconductor Devices

🎯 Learning Objectives

Understand characteristics and applications of power semiconductors
Analyze power switching device performance parameters
Compare different power device technologies
Design power conversion circuits using appropriate devices

🔧 Power Electronics & Renewable Energy Systems

This module covers the intersection of power electronics and renewable energy systems, exploring how advanced semiconductor technologies enable sustainable power generation and grid integration.

1.1 Introduction to Power Semiconductors

Power semiconductors are the fundamental building blocks of modern power electronic systems. Unlike signal-level semiconductors, these devices must handle high currents, voltages, and power levels while maintaining efficient operation and reliability.

Power Electronics Inverter System showing IGBT modules and heat sinks

Figure 1.1: Advanced power electronics inverter system with IGBT modules for renewable energy applications

🔌 Key Power Device Categories

Controlled Switches

IGBTs - Insulated Gate Bipolar Transistors
MOSFETs - Metal-Oxide-Semiconductor FETs
SiC - Silicon Carbide devices
GaN - Gallium Nitride devices

Controlled Rectifiers

SCRs - Silicon Controlled Rectifiers
TRIACs - Triode AC switches
GTOs - Gate Turn-Off thyristors
MCTs - MOS-Controlled Thyristors

1.2 Power MOSFET Characteristics

Power MOSFETs are widely used for their high switching speed and low on-state resistance, particularly in DC-DC converters and motor drive applications.

📐 Key MOSFET Equations

On-State Resistance:

$$R_{DS(on)} = \frac{L}{\mu_n C_{ox} W (V_{GS} - V_{TH})}$$

Gate Charge:

$$Q_g = C_{iss} V_{GS} = (C_{gd} + C_{gs}) V_{GS}$$

Switching Energy:

$$E_{sw} = \frac{1}{2} V_{DS} I_D (t_{on} + t_{off})$$

📊 MOSFET Characteristics Comparison

1.3 IGBT Technology

Insulated Gate Bipolar Transistors (IGBTs) combine the high input impedance of MOSFETs with the low conduction losses of bipolar transistors, making them ideal for high-power applications.

⚡ MOSFET vs IGBT Performance

Parameter	Power MOSFET	IGBT	Typical Application
Voltage Rating	Up to 200V	Up to 6000V	Switching applications
Switching Speed	Very Fast (ns)	Fast (µs)	High-frequency operation
On-State Drop	Higher at high current	Lower at high current	High-power applications
Thermal Resistance	Low	Moderate	Heat dissipation
Gate Drive	Simple	Simple	Control complexity

⚠️ IGBT Design Considerations

Latch-up Protection: IGBTs can experience latch-up in certain conditions
Turn-off Safe Operating Area (RBSOA): Critical for reliable operation
Gate Drive Requirements: Need negative gate voltage for turn-off
Temperature Monitoring: Junction temperature is critical parameter

1.4 Wide Bandgap Semiconductors (SiC & GaN)

Silicon Carbide (SiC) and Gallium Nitride (GaN) semiconductors offer superior performance compared to silicon devices, enabling higher efficiency and power density.

🔬 Semiconductor Material Properties

Property	Silicon (Si)	SiC	GaN	Unit
Bandgap	1.12	3.26	3.40	eV
Breakdown Field	0.3	3.0	3.3	MV/cm
Thermal Conductivity	150	490	130	W/m·K
Electron Mobility	1500	1000	2000	cm²/V·s
Operating Temperature	150	600	400	°C

✅ SiC Advantages

10x higher breakdown voltage
3x higher thermal conductivity
Higher switching frequency
Better thermal management
Higher operating temperatures

✅ GaN Advantages

Highest switching speed
Lower losses at high frequency
Higher efficiency
Compact size possible
Lower parasitic capacitance

🔧 Interactive Device Selector

Application:

Power Level:

Switching Frequency:

📝 Example: Power Device Selection for Solar Inverter

Application: 10kW Grid-tied Solar Inverter

Requirements:

Operating Voltage: 600V DC
Grid Frequency: 60Hz
Efficiency Target: >98%
Switching Frequency: 16kHz

🔧 Design Solution

Step 1: Voltage Analysis

V_DS Rating = V_DC × Safety Factor = 600V × 1.3 = 780V

→ Minimum device voltage rating: 800V

Step 2: Current Analysis

I_AC = P / (√3 × V_AC) = 10000 / (√3 × 480) = 12A

I_DC = I_AC × √2 = 12A × √2 = 17A

→ Include surge factor: 17A × 2.5 = 42.5A

Step 3: Device Selection

Recommended: 1200V SiC MOSFET (or 650V SiC MOSFET with proper rating)

Rating: >50A, R_DS(on) < 80mΩ

Step 4: Switching Frequency Consideration

16kHz operation requires fast-switching device

SiC MOSFET ideal due to low switching losses

2. DC-DC Power Converters

🎯 Learning Objectives

Analyze operating principles of different DC-DC converter topologies
Design buck, boost, and buck-boost converters
Calculate converter efficiency and thermal requirements
Implement control strategies for DC-DC converters

2.1 Buck (Step-Down) Converters

The buck converter is the most fundamental DC-DC topology, providing efficient voltage reduction from a higher voltage source to a lower voltage load.

⚡ Buck Converter Circuit Analysis

📐 Buck Converter Analysis

Voltage Conversion Ratio:

$$\frac{V_{out}}{V_{in}} = D$$

Inductor Current:

$$I_L = \frac{V_{out}}{R_{load}}$$

Inductor Ripple Current:

$$\Delta I_L = \frac{V_{in} \cdot D \cdot (1-D)}{f_s \cdot L}$$

Output Voltage Ripple:

$$\Delta V_{out} = \frac{\Delta I_L}{8 \cdot f_s \cdot C}$$

📈 Buck Converter Operating Waveforms

🔧 Buck Converter Design Procedure

Step 1: Specification

V_in = 12V, V_out = 5V, I_out = 2A, η = 90%

Step 2: Duty Cycle

D = V_out / V_in = 5/12 = 0.417

Step 3: Switch Selection

I_switch = I_out = 2A

V_switch ≥ V_in × 1.5 = 18V

→ Choose: 40V, 5A MOSFET

Step 4: Inductor Selection

Assume f_s = 100kHz, ΔI_L = 20% of I_out

L = V_in × D / (f_s × ΔI_L) = 12 × 0.417 / (100k × 0.4) = 125µH

Step 5: Capacitor Selection

Assume ΔV_out = 1% of V_out = 50mV

C = ΔI_L / (8 × f_s × ΔV_out) = 0.4 / (8 × 100k × 0.05) = 100µF

2.2 Boost (Step-Up) Converters

Boost converters increase the output voltage above the input voltage, making them essential for battery-powered applications and renewable energy systems.

🔋 Continuous Conduction Mode (CCM)

Inductor current never reaches zero during operation

Condition:

$$L > \frac{(1-D)^2 R}{2 f_s}$$

✅ Advantages:

Lower output ripple
Better control characteristics
Higher efficiency

⚡ Discontinuous Conduction Mode (DCM)

Inductor current reaches zero during some switching period

Condition:

$$L < \frac{(1-D)^2 R}{2 f_s}$$

✅ Advantages:

Simpler control
Lower component count
Better light-load efficiency

📐 Boost Converter Analysis

Voltage Conversion Ratio (CCM):

$$\frac{V_{out}}{V_{in}} = \frac{1}{1-D}$$

Voltage Conversion Ratio (DCM):

$$\frac{V_{out}}{V_{in}} = \sqrt{\frac{D^2 R}{2 L f_s}}$$

Average Inductor Current:

$$I_L = \frac{I_{out}}{1-D}$$

Peak Switch Current:

$$I_{switch(peak)} = I_L + \frac{\Delta I_L}{2}$$

2.3 Buck-Boost Converters

Buck-boost converters provide both step-down and step-up functionality, allowing output voltages greater than or less than the input voltage.

🔄 Topologies Comparison

Topology	Input-Output Relation	Polarity	Switch Count	Efficiency
Buck	V_out = D × V_in	Positive	1	High (85-95%)
Boost	V_out = V_in / (1-D)	Positive	1	High (85-95%)
Buck-Boost	V_out = D/(1-D) × V_in	Inverted	1	Medium (80-90%)
SEPIC	V_out = D/(1-D) × V_in	Non-inverted	1	Medium (75-85%)

💡 Application Examples

Buck Converters:

Computer power supplies
LED drivers
Battery charging circuits
Motor speed control

Boost Converters:

Solar cell charging
Battery backup systems
RF power amplifiers
Portable electronics

🛠️ Interactive Buck Converter Designer

Design Inputs

Input Voltage (V):

Output Voltage (V):

Output Current (A):

Switching Frequency (kHz):

Target Ripple (%):

Design Results

Duty Cycle: -

Inductance: -

Capacitance: -

Inductor Current: -

Switch Rating: -

3. AC-DC and DC-AC Converters

🎯 Learning Objectives

Understand operation principles of rectifiers and inverters
Analyze rectifier topologies and their characteristics
Design inverter circuits for AC power conversion
Implement PWM techniques for power converters

3.1 AC-DC Rectifiers

Rectifiers convert AC power to DC power and are fundamental components in power supplies, motor drives, and renewable energy systems.

⚡ Rectifier Topologies

Half-Wave Rectifier

Single diode
Poor utilization
High ripple
Simple construction

Peak Output: Vpk/π

Average Output: Vpk/π

Ripple Factor: 1.21

Full-Wave Rectifier

Four diodes (bridge)
Better utilization
Lower ripple
Center-tap option

Peak Output: 2Vpk/π

Average Output: 2Vpk/π

Ripple Factor: 0.48

Active Rectifier

Controlled switches
Power factor correction
Bidirectional operation
High efficiency

Power Factor: >0.99

THD: <5%

Efficiency: >98%

📐 Rectifier Analysis

Full-Wave Bridge Rectifier:

Average Output Voltage:

$$V_{dc} = \frac{2V_m}{\pi}$$

Peak Inverse Voltage:

$$V_{piv} = V_m$$

Rectification Efficiency:

$$\eta = \frac{0.406}{1 + \frac{R_f}{R_L}}$$

Ripple Factor:

$$\gamma = \frac{V_{r(rms)}}{V_{dc}} = \frac{1}{4\sqrt{3} f R_L C}$$

🔄 Active Rectifier Operation

Active rectifiers use controlled switches to shape the input current waveform, achieving near-unity power factor and reduced harmonic content.

🎯 Control Strategy

Current Control Loop:

Regulates input current to follow sinusoidal reference

$$I_{ctrl} = K_p(e_{curr}) + K_i\int e_{curr} dt$$

Voltage Control Loop:

Regulates DC output voltage to reference value

$$I_{ref} = K_p(e_{volt}) + K_i\int e_{volt} dt$$

PWM Generation:

Digital control generates switching signals

$$Duty_{cycle} = \frac{I_{ctrl}}{I_{ref}} \sin(\omega t)$$

3.2 DC-AC Inverters

Inverters convert DC power to AC power and are essential for renewable energy systems, UPS systems, and motor drives.

⚡ Inverter Topologies

Half-Bridge Inverter

Configuration: Two switches and two capacitors

Output: Half-cycle switching

Applications: Low-power applications

✅ Advantages:

Simple structure
Low component count
Cost-effective

❌ Disadvantages:

Requires floating load
Higher harmonic content
Limited power rating

📊 Performance Metrics

Fundamental Output	$V_1 = \frac{V_{dc}}{2}$
THD	~80%
Efficiency	85-90%
Power Factor	0.7-0.8

Full-Bridge (H-Bridge) Inverter

Configuration: Four switches in bridge configuration

Output: Full-cycle switching

Applications: High-power applications

✅ Advantages:

No floating load required
Lower harmonic content
Higher power capability
Flexible control

❌ Disadvantages:

Higher component count
More complex control
Higher cost

📊 Performance Metrics

Fundamental Output	$V_1 = V_{dc}$
THD (with PWM)	<5%
Efficiency	90-95%
Power Factor	>0.95

Three-Phase Inverter

Configuration: Six switches for three-phase output

Output: Three-phase AC power

Applications: Motor drives, grid-tied systems

✅ Advantages:

Three-phase output
Higher power density
Lower ripple current
Better utilization

❌ Disadvantages:

Most complex topology
Highest component count
Complex switching control

📊 Performance Metrics

Line-to-Line Voltage	$V_LL = \frac{\sqrt{3}}{2}V_{dc}$
THD (with PWM)	<3%
Efficiency	92-96%
Power Factor	>0.98

3.3 PWM Techniques

Pulse Width Modulation (PWM) techniques control the output voltage and frequency while minimizing harmonic content and improving power quality.

📡 PWM Modulation Techniques

Sine PWM (SPWM)

Compares sinusoidal reference with high-frequency triangular carrier to generate PWM signals.

Modulation Index:

$$m_a = \frac{V_{ref}}{V_{carrier}}$$

Fundamental Frequency:

$$f_{fundamental} = f_{mod}$$

Carrier Frequency:

$$f_{carrier} = n \times f_{mod}$$

✅ Advantages:

Simple implementation
Low harmonic distortion
Easy control algorithm

Space Vector PWM (SVPWM)

Uses space vector representation to achieve better DC bus utilization and lower switching losses.

Space Vector:

$$\vec{V} = \frac{2}{3}(V_a + aV_b + a^2V_c)$$

Modulation Index:

$$m_a = \frac{V_{1max}}{V_{DC}/\sqrt{2}} = \frac{2V_{1max}}{V_{DC}/\sqrt{2}} = \frac{2\sqrt{2}V_{1max}}{V_{DC}}$$

✅ Advantages:

15% higher DC utilization
Lower switching losses
Better output quality

Selective Harmonic Elimination (SHE)

Eliminates specific harmonic components by solving equations for switching angles.

Harmonic Elimination:

$$\sum_{k=1}^{N/2} M_k \sin(k\theta_k) = V_1$$

Fundamental Component:

$$V_1 = \frac{4V_{dc}}{\pi} \sum_{k=1}^{N/2} M_k \cos(k\theta_k)$$

✅ Advantages:

Specific harmonic control
Minimum switching frequency
Optimal harmonic performance

📊 PWM Technique Comparison

3.4 Matrix Converters

Matrix converters provide direct AC-AC conversion without intermediate DC storage, offering advantages in terms of size, weight, and efficiency.

🔄 Direct Conversion

No intermediate DC link required

Direct AC-AC power conversion
Bidirectional power flow capability
Regenerative operation support
Reduced component count

⚡ Performance Benefits

Superior electrical characteristics

Near-unity power factor
Low harmonic distortion
High efficiency (>95%)
Compact design

📐 Matrix Converter Analysis

Output Voltage:

$$\begin{bmatrix} V_o \\ I_i \end{bmatrix} = \begin{bmatrix} 0 & T \\ T^T & 0 \end{bmatrix} \begin{bmatrix} V_i \\ I_o \end{bmatrix}$$

Transfer Matrix:

$$T = \begin{bmatrix} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{bmatrix}$$

Voltage Constraint:

$$\sum_{j=1}^{3} T_{ij} = \frac{1}{3}$$

Current Constraint:

$$\sum_{i=1}^{3} T_{ij} = 0$$

📝 Example: Three-Phase Inverter Design

Specification: 10kW, 480V, 60Hz Three-Phase Inverter

🔧 Design Parameters

Parameter	Value	Calculation
DC Bus Voltage	650V	V_DC = V_LL × √2 = 480 × √2
Rated Current	12A	I = P/(√3 × V) = 10000/(√3 × 480)
Switching Frequency	16kHz	Audio frequency above audibility
Modulation Index	0.9	Typical for good utilization

⚡ Component Selection

IGBT Selection:

Voltage Rating: >1200V (V_DC × 1.5)
Current Rating: >25A (I_line × 1.5 × √2)
Switching Frequency: >20kHz
Package: Isolated module

Snubber Circuit:

Snubber Capacitor: 10nF
Snubber Resistor: 100Ω
RCD Snubber Configuration
Turn-on losses minimization

4. Solar Photovoltaic Systems

🎯 Learning Objectives

Understand PV cell operation and characteristics
Design solar panel arrays and connections
Implement maximum power point tracking (MPPT)
Calculate system sizing and energy yield

4.1 PV Cell Technology and Characteristics

Photovoltaic cells convert sunlight directly into electricity through the photovoltaic effect. The efficiency and performance characteristics depend on cell material, technology, and operating conditions.

Solar Photovoltaic System showing panels, inverter, and grid connection

Figure 4.1: Complete solar PV system with panels, inverter, and grid connection infrastructure

📐 PV Cell Equations

Cell Current:

$$I = I_{ph} - I_s \left[e^{\frac{q(V + IR_s)}{nkT}} - 1\right] - \frac{V + IR_s}{R_{sh}}$$

Maximum Power Point:

$$P_{mpp} = V_{mpp} \times I_{mpp}$$

Cell Efficiency:

$$\eta = \frac{P_{mpp}}{P_{in}} = \frac{V_{oc} \times I_{sc} \times FF}{P_{in}}$$

4.2 MPPT Algorithms

Maximum Power Point Tracking ensures PV systems operate at their optimal power output under varying environmental conditions.

📊 Perturb and Observe (P&O)

Most common MPPT algorithm with good performance

✅ Advantages:

Simple implementation
Low cost
Good performance in uniform conditions

📈 Incremental Conductance

More sophisticated algorithm with faster tracking

✅ Advantages:

No oscillations at maximum power point
Better performance in rapidly changing conditions
Fast tracking speed

☀️ Solar PV System Sizing Calculator

Daily Energy Demand (kWh):

Peak Sun Hours:

PV Module Efficiency (%):

System Losses (%):

Battery Days of Autonomy:

Battery Efficiency (%):

📊 PV System Design Results

Enter system parameters and click "Calculate PV System" to see results.

5. Wind Energy Conversion Systems

🎯 Learning Objectives

Understand wind turbine operation and aerodynamics
Analyze wind power extraction principles
Design wind energy conversion systems
Evaluate power coefficient and turbine performance

5.1 Wind Turbine Aerodynamics

Wind turbines convert the kinetic energy of moving air into electrical energy through aerodynamic principles and power electronic conversion.

Wind turbine generator system with electrical components

Figure 5.1: Wind turbine generator system showing the electrical conversion chain from mechanical to electrical power

📐 Wind Power Equations

Available Wind Power:

$$P_{wind} = \frac{1}{2} \rho A v^3$$

Power Extracted by Turbine:

$$P_{turbine} = \frac{1}{2} \rho A v^3 C_p$$

Tip Speed Ratio:

$$\lambda = \frac{\omega R}{v}$$

Bet Limit (Maximum Cp):

$$C_{p,max} = \frac{16}{27} \approx 0.593$$

5.2 Wind Turbine Generator Systems

Modern wind turbines use various generator technologies, with doubly-fed induction generators (DFIG) and permanent magnet synchronous generators (PMSG) being most common.

⚡ DFIG (Doubly-Fed Induction Generator)

Variable speed operation with partial power converter

Rated power: 1.5-5 MW typical
Converter rating: ~30% of generator rating
Speed range: ±30% synchronous speed
Cost effective for large systems

🔄 PMSG (Permanent Magnet Synchronous Generator)

Direct drive system with full power converter

Rated power: 0.5-10 MW typical
Converter rating: 100% of generator rating
Gearbox elimination (direct drive)
Higher efficiency, lower maintenance

💨 Wind Energy Calculation Tool

Wind Speed (m/s):

Rotor Diameter (m):

Power Coefficient (Cp):

Air Density (kg/m³):

Generator Efficiency (%):

Gearbox Efficiency (%):

🌪️ Wind Power Analysis Results

Enter wind and turbine parameters to calculate power output.

6. Energy Storage Systems

🎯 Learning Objectives

Understand different energy storage technologies
Analyze battery systems for renewable energy
Design energy storage for grid applications
Evaluate storage system economics and performance

6.1 Battery Energy Storage Systems

Energy storage systems are crucial for renewable energy integration, providing grid stability, peak shaving, and backup power capabilities.

Energy storage system with batteries and power electronics

Figure 6.1: Modern battery energy storage system showing battery racks and power conversion equipment

📐 Energy Storage Equations

Energy Capacity:

$$E = V_{nominal} \times C_{rated} \times DOD$$

Power Rating:

$$P = V_{nominal} \times I_{max}$$

State of Charge (SOC):

$$SOC = \frac{C_{remaining}}{C_{capacity}} \times 100\%$$

Round-trip Efficiency:

$$\eta_{round-trip} = \frac{E_{discharge}}{E_{charge}} \times 100\%$$

6.2 Storage Technologies Comparison

Different storage technologies offer varying characteristics in terms of power density, energy density, response time, and cycle life.

Technology	Energy Density (Wh/kg)	Power Density (W/kg)	Cycle Life	Response Time	Cost ($/kWh)
Lithium-ion	100-265	1,000-10,000	1,000-10,000	Milliseconds	200-400
Lead-acid	30-50	100-500	500-2,000	Seconds	150-300
Flow Battery	20-70	10-100	10,000-20,000	Seconds	250-500
Supercapacitor	5-10	10,000-100,000	100,000-1,000,000	Microseconds	2,000-10,000

🔋 Energy Storage System Designer

Required Energy (kWh):

Required Power (kW):

Battery Technology:

Depth of Discharge (%):

System Efficiency (%):

Cost per kWh ($):

⚡ Energy Storage Design Results

Enter storage requirements to get system design recommendations.

7. Smart Grid and Grid Integration Technologies

🎯 Learning Objectives

Understand smart grid architecture and communication
Analyze grid integration challenges and solutions
Implement grid codes and power quality standards
Design microgrid and islanding systems

7.1 Grid Integration Technologies

Successful integration of renewable energy systems requires advanced power electronics, communication technologies, and grid management systems to ensure stability and reliability.

Smart grid network showing renewable energy integration

Figure 7.1: Smart grid infrastructure showing renewable energy sources, storage, and grid integration points

📐 Grid Integration Parameters

Grid Synchronization:

$$\Delta f = f_{measured} - f_{reference}$$

Voltage Regulation:

$$\Delta V = \frac{V_{actual} - V_{reference}}{V_{reference}} \times 100\%$$

Harmonic Distortion:

$$THD = \frac{\sqrt{\sum_{h=2}^{\infty} V_h^2}}{V_1} \times 100\%$$

Power Factor:

$$PF = \frac{P}{S} = \cos\theta$$

7.2 Grid Code Requirements

Modern grid codes specify technical requirements for renewable energy systems to ensure grid stability and power quality.

⚡ Frequency Response

Frequency support: ±0.5 Hz typical range
Active power vs frequency droop
Fast frequency response capability
Grid forming capability for weak grids

🔌 Voltage Support

Reactive power capability: ±0.95 PF
Voltage regulation: ±3% typical
Fault ride-through capability
Harmonic current limits: THD < 5%

🏗️ Grid Integration Assessment Tool

RE Generation (MW):

Grid Short Circuit Ratio:

Grid Frequency (Hz):

Grid Voltage (kV):

Power Factor:

Total Harmonic Distortion (%):

🔗 Grid Integration Assessment

Enter system parameters to assess grid integration requirements.

📝 Module Assessment

Test your understanding of Power Electronics & Renewable Energy Systems with this comprehensive assessment.

Question 1: Power MOSFET Selection

A 2kW DC-DC converter requires an input voltage of 48V and output voltage of 12V. The switching frequency is 50kHz. Select the appropriate power MOSFET rating.

60V, 50A MOSFET 100V, 25A MOSFET 80V, 40A MOSFET 120V, 30A MOSFET

Correct Answer: b

For 2kW at 12V, output current = 166.7A. Input current = 41.7A. With ripple and safety factors, a 100V, 25A MOSFET would be undersized for continuous operation. The correct choice should handle at least 70A continuous current and >75V voltage rating.

Question 2: Buck Converter Design

Design a buck converter with V_in = 24V, V_out = 12V, I_out = 5A, f_s = 100kHz, and target inductor ripple current of 20%. Calculate the required inductance.

Given:

V_in = 24V, V_out = 12V, I_out = 5A
f_s = 100kHz, ΔI_L = 20% of I_out = 1A
Duty cycle D = V_out/V_in = 12/24 = 0.5

Solution:

L = (V_in × D × (1-D)) / (f_s × ΔI_L)

L = (24 × 0.5 × 0.5) / (100k × 1) = 60µH

Question 3: Inverter THD Analysis

A three-phase inverter produces a fundamental component of 480V RMS at 60Hz. The 5th harmonic amplitude is 96V RMS, and the 7th harmonic amplitude is 68.6V RMS. Calculate the Total Harmonic Distortion (THD).

Solution:

THD = √(V₅² + V₇²) / V₁

THD = √(96² + 68.6²) / 480

THD = √(9216 + 4705.96) / 480

THD = √13921.96 / 480 = 118.03 / 480 = 24.6%

Question 4: Solar PV System Sizing

A residential solar PV system must generate 15kWh/day in a location with 5 sun-hours. The panel efficiency is 18%, and system losses are 15%. Calculate the required solar panel array size.

Solution:

Energy Required: 15kWh/day

Adjusted for losses: 15 / (1 - 0.15) = 17.65kWh/day

Peak Power Required: 17.65kWh / 5sun-hours = 3.53kW

Panel Array Size: 3.53kW / 0.18 = 19.6kW

Therefore, approximately 20kW of solar panels required