Aux 2 Unit 6 Q3

True power, also referred to as active power or real power, is the actual power consumed by a load to perform useful work. In electrical circuits, it’s the power that is converted into other forms of energy like heat, light, or mechanical motion.

Key Points about True Power:

• Measured in Watts (W): The unit of measurement for true power is watts, symbolized by the letter ‘P’.
• AC Circuits: In AC circuits, true power is the portion of the apparent power that actually does work. It’s calculated by considering the phase angle between voltage and current waveforms.
• Formula:P = V * I * cos(θ), where:
  • P = True Power (watts)
  • V = Voltage (volts)
  • I = Current (amperes)
  • θ = Phase angle between voltage and current
• Power Factor: The cosine of the phase angle (cos θ) is called the power factor. It represents the efficiency of power usage in an AC circuit. A power factor of 1 means all the apparent power is being converted into true power (ideal scenario), while a lower power factor indicates some power is being wasted as reactive power.

Importance of True Power:

• Billing: Electricity bills are typically based on the true power consumed, not the apparent power.
• System Design: Electrical systems are designed to handle the apparent power, but the true power determines the actual load and capacity requirements.
• Efficiency: A high power factor indicates efficient power usage, while a low power factor means energy is wasted and additional capacity might be needed in the system.

Example:

• A motor draws 10 amps at 220 volts with a power factor of 0.8.
• Apparent power (S) = V * I = 220 V * 10 A = 2200 VA
• True power (P) = V * I * cos(θ) = 220 V * 10 A * 0.8 = 1760 W

In this example, the motor is actually using 1760 watts of true power to do work, while the remaining 440 VA is reactive power that’s not performing useful work but still flows in the circuit.

In conclusion, true power is the actual power that does work in an electrical circuit. It’s the power we pay for and the power that determines the actual load on the system. Understanding true power and power factor is crucial for efficient energy management and electrical system design.

Apparent power, in the context of electrical circuits, represents the total power that appears to be flowing in an AC circuit. It’s the product of the voltage and current in the circuit, without considering the phase relationship between them.

Key points about apparent power:

• Measured in Volt-Amperes (VA): The unit of measurement for apparent power is volt-amperes, symbolized by the letter ‘S’.
• AC Circuits: In AC circuits, apparent power consists of two components:
  • True Power (P): The actual power consumed by the load to perform useful work (measured in Watts).
  • Reactive Power (Q): The power that oscillates back and forth between the source and the load due to the presence of reactive components like inductors and capacitors (measured in Volt-Amperes Reactive or VAR).  
• Formula:S = V * I, where:
  • S = Apparent Power (volt-amperes)
  • V = Voltage (volts)
  • I = Current (amperes)
• Relationship with True Power: S² = P² + Q² (from the Power Triangle)

Why is apparent power important?

• System Design: Electrical systems, including generators, transformers, and cables, need to be sized to handle the apparent power, not just the true power, to avoid overloading and ensure safe operation.
• Power Factor: The ratio of true power to apparent power is called the power factor (PF = P/S). A low power factor indicates inefficient power usage, with a larger portion of the apparent power being wasted as reactive power. This can lead to increased energy costs and the need for larger electrical infrastructure.

Example:

• A motor draws 10 amps at 220 volts with a power factor of 0.8.
• Apparent power (S) = V * I = 220 V * 10 A = 2200 VA
• True power (P) = V * I * cos(θ) = 220 V * 10 A * 0.8 = 1760 W
• Reactive power (Q) = √(S² – P²) = √(2200² – 1760²) = 1320 VAR

In this example, the motor’s apparent power is 2200 VA, but only 1760 W is actually doing useful work (true power). The remaining 1320 VAR is reactive power, which is necessary for the motor’s operation but doesn’t directly contribute to the work being done.

In conclusion, apparent power is a crucial concept in AC circuits. It represents the total power flowing in the circuit and is essential for designing and sizing electrical systems. Understanding the relationship between apparent power, true power, and reactive power helps ensure efficient and safe operation of electrical equipment.

Reactive power, in the context of AC electrical circuits, is the portion of power that flows back and forth between the source (like a generator) and the load (like a motor or transformer) without actually being consumed or doing any useful work. It’s associated with the energy stored and released by reactive components in the circuit, mainly inductors and capacitors.

Key points about reactive power:

• Measured in Volt-Amperes Reactive (VAR): It’s represented by the symbol ‘Q’ and measured in VAR.
• Energy Storage and Release: Reactive power represents the energy that’s temporarily stored in the magnetic field of inductors or the electric field of capacitors during one part of the AC cycle and then returned to the source during another part. This back-and-forth flow of energy doesn’t contribute to actual work being done.
• Phase Relationship: It arises due to the phase difference between the voltage and current waveforms in an AC circuit. In a purely resistive circuit, voltage and current are in phase, and there’s no reactive power. But in circuits with inductors or capacitors, the current either lags or leads the voltage, creating reactive power.
• Inductive vs. Capacitive:
  • Inductive loads (motors, transformers): The current lags behind the voltage, resulting in positive reactive power (absorbing VARs).
  • Capacitive loads (power factor correction capacitors): The current leads the voltage, resulting in negative reactive power (supplying VARs).

Why is reactive power important?

• Affects Apparent Power: Even though reactive power doesn’t do useful work, it still contributes to the total apparent power (S) flowing in the circuit, which is the product of voltage and current.
• Impacts Power Factor: The power factor (cos θ) is the ratio of true power (P) to apparent power (S). A low power factor means a significant portion of the apparent power is reactive, leading to inefficiencies in the electrical system.
• Increased Current and Losses: Higher reactive power results in higher currents flowing in the system, even though they are not doing useful work. This leads to increased losses in transmission lines and transformers, requiring larger infrastructure to handle the additional current.
• Voltage Regulation Issues: Reactive power can also cause voltage fluctuations and affect the stability of the power system.

Managing Reactive Power:

• Power Factor Correction: Capacitors can be added to inductive loads to compensate for their reactive power demand, improving the power factor and reducing the overall current flow in the system.
• Efficient Equipment: Using equipment with a high power factor minimizes reactive power requirements.

In summary, reactive power, while not directly contributing to useful work, is an essential aspect of AC circuits that needs to be understood and managed to ensure efficient and stable operation of the electrical system.

In AC electrical circuits, power factor (PF) is a measure of how effectively electrical power is being used. It’s the ratio of the real power (the power that actually does useful work) to the apparent power (the total power supplied to the circuit).

Mathematically, it’s represented as:

• Power Factor (PF) = Real Power (Watts) / Apparent Power (Volt-Amperes)

Or, it can also be expressed as:

• PF = cos(θ), where θ is the phase angle between the voltage and current waveforms.

Key points about power factor:

• Range: Power factor ranges from 0 to 1.
• Ideal Value: A power factor of 1 (or 100%) is ideal, meaning all the supplied power is being used effectively.
• Low Power Factor: A low power factor (less than 0.9 or 90%) indicates that a significant portion of the power is reactive power, which doesn’t do useful work but still flows in the circuit, causing inefficiencies.
• Causes of Low Power Factor: Inductive loads like motors and transformers are the primary cause of low power factor, as they require reactive power to create magnetic fields.
• Effects of Low Power Factor:
  • Increased current flow in the circuit, leading to higher losses and the need for larger cables and equipment.
  • Voltage drops and potential instability in the power system.
  • Increased electricity bills, as utilities often penalize for low power factor.

Improving Power Factor:

• Power Factor Correction: Adding capacitors to the circuit can compensate for the reactive power demand of inductive loads, bringing the power factor closer to 1.
• Using Efficient Equipment: Choosing equipment with a high power factor rating helps minimize reactive power requirements.

In summary, power factor is a crucial measure of how efficiently electrical power is being utilized in an AC circuit. A high power factor is desirable as it indicates minimal wastage of power and reduces the burden on the electrical system.

Certainly, let’s explore the relationship between true power (kW), apparent power (kVA), reactive power (kVAr), and power factor in electrical generation.

Understanding the Concepts:

• True Power (kW): This is the actual power consumed by a load to perform useful work. It’s the power converted into other forms of energy, like heat, light, or mechanical motion.
• Apparent Power (kVA): The total power that appears to be flowing in an AC circuit. It’s the product of the voltage and current, regardless of their phase relationship.
• Reactive Power (kVAr): The power that oscillates back and forth between the source and the load due to the presence of reactive components (inductors and capacitors). It doesn’t perform useful work but is necessary for the operation of certain equipment.
• Power Factor: The ratio of true power to apparent power (PF = kW / kVA). It represents how efficiently the electrical power is being used. A power factor of 1 (or 100%) means all the apparent power is converted into true power (ideal scenario).

Relationship:

These four quantities are related through the power triangle, a right-angled triangle where:

• Hypotenuse: Represents the apparent power (kVA).
• Adjacent side: Represents the true power (kW).
• Opposite side: Represents the reactive power (kVAr).
• Angle: The angle between the hypotenuse and the adjacent side is the phase angle (θ), and its cosine (cos θ) is the power factor.

Mathematical Expression:

• S² = P² + Q²
  • Where:
    • S = Apparent power (kVA)
    • P = True power (kW)
    • Q = Reactive power (kVAr)
• PF = P / S = cos θ

Significance in Electrical Generation:

• Generator Sizing: Generators are rated in kVA (apparent power) because they need to be able to handle both the true power and the reactive power demands of the load.
• Efficiency and Losses: A low power factor means a higher proportion of reactive power, leading to increased current flow, losses in transmission lines, and inefficient utilization of the generator’s capacity.
• Power Factor Correction: To improve efficiency and reduce losses, power factor correction techniques are used to minimize reactive power. This typically involves adding capacitors to the system to compensate for the inductive loads (like motors) that cause lagging power factor.
• Billing: While electricity bills for residential consumers are usually based on true power (kWh), industrial users might also be charged for reactive power (kVArh) or penalized for low power factor.

In conclusion:

Understanding the relationship between true power, apparent power, reactive power, and power factor is crucial in electrical generation. It helps optimize system design, improve efficiency, reduce losses, and ensure the reliable and cost-effective delivery of electrical power.

When two generators are connected in parallel, they share the responsibility of supplying both the true power (kW) and the reactive power (kVAr) demand of the connected loads. Achieving proper load sharing is essential for efficient operation and preventing overload or instability in the system.

Sharing of True Power (kW):

• Governor Control: The primary mechanism for controlling the sharing of true power is the governor system of each generator’s prime mover (typically a diesel engine or turbine).
• Speed Droop: The governors are set with a slight speed droop characteristic, meaning that as the load on a generator increases, its speed decreases slightly.
• Load Sharing: When two generators are in parallel, a slight difference in their speeds will cause the generator with the higher speed (lower load) to take on a larger share of the true power demand. This self-adjusting mechanism helps balance the load between the generators.

Sharing of Reactive Power (kVAr):

• AVR Control: The sharing of reactive power is primarily controlled by the Automatic Voltage Regulator (AVR) of each generator.
• Excitation Control: The AVR adjusts the field excitation of the generator, which in turn affects its reactive power output.
• Load Sharing: When two generators are in parallel, a slight difference in their terminal voltages will cause the generator with the higher voltage to supply more reactive power. By adjusting the excitation of each generator, the reactive power sharing can be balanced.

Key Points:

• Proportional Sharing: Ideally, the true power and reactive power should be shared proportionally between the generators based on their respective capacities.
• Power Factor: The power factor of each generator will vary depending on its share of the reactive power. A generator supplying more reactive power will have a lower power factor.
• Manual Adjustment: In some cases, manual adjustments to the governor or AVR settings may be necessary to fine-tune the load sharing and ensure optimal operation.
• Load Changes: When the overall load on the system changes, both the true power and reactive power sharing will automatically adjust to maintain balance between the generators.
• Importance of Proper Sharing: Improper load sharing can lead to overloading of one generator, reduced efficiency, and potential instability in the system.

In summary, the sharing of true power and reactive power between two generators connected in parallel is achieved through a combination of governor control (for true power) and AVR control (for reactive power). These control systems work in tandem to automatically adjust the output of each generator, ensuring a balanced and efficient load sharing.

Additional Notes:

• It’s essential to ensure that the generators are properly synchronized before connecting them in parallel. This involves matching their voltage, frequency, and phase sequence.
• Modern power management systems often incorporate sophisticated control algorithms to optimize load sharing and ensure stable operation of paralleled generators.
• Regular monitoring and maintenance of the governor and AVR systems are crucial for maintaining proper load sharing and preventing issues in the parallel operation.

Understanding these principles helps ensure efficient and reliable power generation when multiple generators are operating in parallel.

Achieving kW and kVAr Load Sharing Stability

Load sharing stability in a paralleled generator system ensures that each generator contributes its fair share of both real power (kW) and reactive power (kVAr) to the overall load. This balance is vital for preventing overload on any single generator, ensuring efficient operation, and maintaining system stability.

Here’s how this stability is achieved:

1. kW (Real Power) Load Sharing:

• Governor Control: The primary mechanism for kW load sharing is the governor on each generator’s prime mover (engine or turbine). The governor adjusts the fuel input to the prime mover to control its speed.
• Speed Droop: Governors are set with a slight “droop” characteristic, meaning that as the load on a generator increases, its speed decreases slightly. This inherent characteristic creates a natural load-sharing mechanism.
• Automatic Adjustment: When the overall load increases, the system frequency tends to drop slightly. The generator with the higher speed (and hence, lower load) will experience a larger speed drop, causing its governor to increase fuel input, thereby taking on a greater share of the increased kW load. This self-adjustment continues until the system reaches a new equilibrium where the load is shared proportionally between the generators based on their capacity.

2. kVAr (Reactive Power) Load Sharing:

• AVR Control: Reactive power sharing is primarily controlled by the Automatic Voltage Regulator (AVR) on each generator. The AVR adjusts the generator’s field excitation, which directly influences its terminal voltage and reactive power output.
• Voltage Droop: Similar to speed droop for kW sharing, AVRs can be set with a slight “voltage droop” characteristic. As the reactive power load on a generator increases, its terminal voltage decreases slightly.
• Automatic Adjustment: If the overall reactive power demand increases, the system voltage tends to drop. The generator with the higher terminal voltage (lower reactive power load) will experience a larger voltage drop, causing its AVR to increase excitation and thus take on a greater share of the increased kVAr load.

Additional Factors and Considerations:

• Generator Ratings: Load sharing is typically done proportionally based on the kW and kVAr ratings of each generator.
• Cross-Current Compensation: To avoid circulating currents between generators (which can lead to instability and losses), cross-current compensation may be employed. This involves adjusting the AVR settings to ensure that the reactive power sharing is balanced even in the absence of significant voltage droop.
• Isochronous Control: Some systems use isochronous control, where the governors maintain a constant speed (and hence frequency) regardless of load changes. In these systems, load sharing is achieved through more complex control algorithms that actively monitor and adjust the fuel input to each generator based on its load and capacity.
• Monitoring and Adjustments: It’s important to continuously monitor the load sharing between generators and make any necessary adjustments to the governor or AVR settings to maintain optimal balance and stability.

Conclusion:

Achieving stable load sharing of both kW and kVAr in a paralleled generator system requires a combination of governor control, AVR control, and potentially additional control mechanisms like cross-current compensation or isochronous control. These systems work together to automatically adjust the output of each generator in response to changing load demands, ensuring efficient and reliable power generation while preventing overload and instability.