This document discusses different techniques for phasor estimation from discrete time signals, including the discrete Fourier transform (DFT). It focuses on explaining the one-cycle DFT method. The one-cycle DFT divides the signal into windows of one cycle and applies the DFT to each window to estimate the phasor. It is shown that with each new sample, the phasor estimate from the DFT advances by 45 degrees. The document also discusses applying the one-cycle DFT to signals with harmonic components and estimating sequence components from three-phase signals.
This document discusses active and reactive power flow control using a Static Synchronous Series Compensator (SSSC). The SSSC injects a controllable voltage in series with a transmission line to regulate power flow. It can control both real and reactive power flow to improve transmission efficiency. The SSSC consists of a voltage source converter connected to the line via a transformer. It provides advantages like power factor correction, load balancing, and reducing harmonic distortion.
Statcom control scheme for power quality improvement of grid connected wind e...Kinnera Kin
This project aims to improve power quality for a grid-connected wind energy system using a STATCOM. The objectives are to maintain unity power factor at the source, meet reactive power needs of the wind generator and non-linear load, and provide fast response using hysteresis current control for the STATCOM. MATLAB/Simulink software is used to simulate the system both with and without STATCOM. The simulation results show that with STATCOM, harmonic distortion is eliminated in the load current and power quality is maintained at the point of common coupling.
power quality improvement in distrution system using D statcom7867867869
The document discusses using a D-STATCOM device to improve power quality issues in distribution systems, such as voltage sags, harmonic distortion, and low power factor. A D-STATCOM injects current into the system using a voltage source converter to regulate voltage and mitigate sags. It can also absorb or generate reactive power to improve power factor and eliminate current harmonics. Simulation results showed the D-STATCOM is effective at mitigating voltage sags and improving power factor when combined with an LCL passive filter.
1. The document discusses a static synchronous series compensator (SSSC), a type of flexible AC transmission system (FACTS) device that controls electric power flow by injecting a controlled voltage in series with a transmission line.
2. The SSSC can provide either capacitive or inductive compensation, depending on whether the injected voltage lags or leads the line current.
3. Digital simulations show that the SSSC can increase or decrease the dynamic power flow in the transmission line depending on the mode of compensation.
This document discusses FACTS (Flexible AC Transmission System) devices. It defines FACTS as using static power electronics controllers to control reactive power and enhance AC transmission system controllability. The document outlines the necessity of FACTS devices to compensate for reactive power and improve power transmission efficiency. It describes different types of FACTS controllers including shunt controllers like STATCOM, TCR, TSR, and TSC. The benefits of FACTS in providing fast, flexible control of transmission parameters and improving power flow capability are also summarized.
Power quality improvement using upqc with soft computing method: Fuzzy logicSakti Prasanna Muduli
Now a days problems regarding power quality is more in large inter connected power systems. There are many method to mitigate these problems but using the latest most efficient compensation method is some what impressive. Here is the brief explanations regarding UPQC using soft computing method(fuzzy logic). This was my academic project along with my friends.
This document discusses active and reactive power flow control using a Static Synchronous Series Compensator (SSSC). The SSSC injects a controllable voltage in series with a transmission line to regulate power flow. It can control both real and reactive power flow to improve transmission efficiency. The SSSC consists of a voltage source converter connected to the line via a transformer. It provides advantages like power factor correction, load balancing, and reducing harmonic distortion.
Statcom control scheme for power quality improvement of grid connected wind e...Kinnera Kin
This project aims to improve power quality for a grid-connected wind energy system using a STATCOM. The objectives are to maintain unity power factor at the source, meet reactive power needs of the wind generator and non-linear load, and provide fast response using hysteresis current control for the STATCOM. MATLAB/Simulink software is used to simulate the system both with and without STATCOM. The simulation results show that with STATCOM, harmonic distortion is eliminated in the load current and power quality is maintained at the point of common coupling.
power quality improvement in distrution system using D statcom7867867869
The document discusses using a D-STATCOM device to improve power quality issues in distribution systems, such as voltage sags, harmonic distortion, and low power factor. A D-STATCOM injects current into the system using a voltage source converter to regulate voltage and mitigate sags. It can also absorb or generate reactive power to improve power factor and eliminate current harmonics. Simulation results showed the D-STATCOM is effective at mitigating voltage sags and improving power factor when combined with an LCL passive filter.
1. The document discusses a static synchronous series compensator (SSSC), a type of flexible AC transmission system (FACTS) device that controls electric power flow by injecting a controlled voltage in series with a transmission line.
2. The SSSC can provide either capacitive or inductive compensation, depending on whether the injected voltage lags or leads the line current.
3. Digital simulations show that the SSSC can increase or decrease the dynamic power flow in the transmission line depending on the mode of compensation.
This document discusses FACTS (Flexible AC Transmission System) devices. It defines FACTS as using static power electronics controllers to control reactive power and enhance AC transmission system controllability. The document outlines the necessity of FACTS devices to compensate for reactive power and improve power transmission efficiency. It describes different types of FACTS controllers including shunt controllers like STATCOM, TCR, TSR, and TSC. The benefits of FACTS in providing fast, flexible control of transmission parameters and improving power flow capability are also summarized.
Power quality improvement using upqc with soft computing method: Fuzzy logicSakti Prasanna Muduli
Now a days problems regarding power quality is more in large inter connected power systems. There are many method to mitigate these problems but using the latest most efficient compensation method is some what impressive. Here is the brief explanations regarding UPQC using soft computing method(fuzzy logic). This was my academic project along with my friends.
The document discusses various objectives and applications of static shunt compensation on transmission lines. Shunt compensation can increase steady-state transmittable power, control voltage profiles, minimize line overvoltage under light loads using shunt reactors, and maintain voltage levels under heavy loads using shunt capacitors. Midpoint shunt compensation significantly increases transmitted power and is best located at the midpoint where voltage sag is maximum. End of line shunt compensation effectively increases voltage stability limits and regulates terminal voltages to prevent voltage instability. Shunt compensation can also improve transient stability and damp power oscillations on transmission lines.
Instantaneous Reactive Power Theory And Its Applicationsarunj89
Instantaneous Reactive Power Theory and its Applications to Active Power Filtering
The document discusses instantaneous reactive power (P-Q) theory, which was introduced by Hirofumi Akagi in 1983. P-Q theory defines instantaneous real and imaginary powers in the time domain, allowing it to be applied to non-sinusoidal systems. It has been widely used for harmonic compensation in active power filters. The document outlines the mathematical basis of P-Q theory, including Clarke transformations, definitions of instantaneous real and imaginary powers, and applications for compensating nonlinear loads. It also discusses developments and applications of P-Q theory, including its use in simulation and compensation of harmonic currents.
NEW STATCOM CONTROL SCHEME FOR POWER QUALITY IMPROVEMENT IN WIND FARM.sannuthi yaramapu
Now a days we are facing so many problems with power quality issues. So in order to mitigate these problems and to improve the power quality we are using new STATCOM control scheme in wind farm.
The document discusses implementing FACTS controllers on a 370km transmission line model to improve performance. It describes FACTS, different types of FACTS controllers including thyristor controlled series compensation (TCSC) and shunt compensation. Simulation results show that with FACTS compensation, the line can transfer more active power with better voltage regulation and power factor compared to the uncompensated line. FACTS increases the line's power transfer capability.
This document discusses smart grids and the role of advanced metering infrastructure in India. It notes that India has one of the weakest electrical grids in the world with high transmission losses. A smart grid uses communication and information technologies to better manage electricity distribution and demand. Advanced metering infrastructure is a key component, allowing two-way communication between utilities and customers to provide energy usage data and enable demand response programs. This can help improve grid reliability and efficiency while empowering consumers.
This document describes a project to improve power quality using a Unified Power Quality Conditioner (UPQC). The UPQC compensates for voltage disturbances and improves current quality using active power filters. It maintains the load voltage despite supply variations. The document outlines the objectives, introduces UPQC components like the shunt and series active power filters, and describes the multivariable controller and Simulink model. The UPQC provides advantages like reduced harmonics, improved waveform quality, and balanced power factor.
This document discusses static shunt compensation on transmission lines. Shunt compensation can increase steady-state transmittable power and control voltage profiles by using shunt reactors to minimize overvoltage under light loads and shunt capacitors to maintain voltage levels under heavy loads. Midpoint shunt compensation regulates voltage along line segments by exchanging only reactive power at the midpoint, significantly increasing transmittable power as the midpoint has the maximum voltage sag. End of line shunt compensation also provides voltage support to prevent instability.
This document discusses the need for transmission interconnections and opportunities provided by FACTS (Flexible AC Transmission Systems) technology. It notes that India has generation surpluses in some grids but deficits in others, and interconnections allow sharing of power to reduce costs. FACTS devices can control power flows and enhance line capacity, enabling more economic energy transfers. They offer advantages over mechanical switching like reduced wear and ability to damp oscillations. FACTS technology opens opportunities to better utilize transmission assets by overcoming thermal, dielectric and stability limitations on line loadings.
Application of Capacitors to Distribution System and Voltage RegulationAmeen San
Application of Capacitors to
Distribution System and Voltage
Regulation
POWER FACTOR IMPROVEMENT,
System Harmonics
Voltage Regulation
Methods of Voltage Control
In microgrid, if fault occurs or any other contingency happens, then the problems would be created which are related to power flow, also there are various protection schemes are used for minimize or eliminate these problems.
Voltage control is used for reactive power balance and P-f control is used for active power control.
Various protection schemes such as, over current protection, differential protection scheme, zoning of network in adaptive protection scheme are used in microgrid system .
The document describes a Simulink model that was created to improve total harmonic distortion (THD) using a shunt active power filter. The model simulates a power system with a non-linear load connected to an ideal grid voltage. The shunt active power filter is connected 0.1 seconds after simulation start and works to compensate for harmonics by producing currents equal in magnitude but opposite in phase to the load harmonics. Simulation results show the THD is reduced from around 30.9% on the load side to 2.79% on the source side once the active filter is connected, below the maximum allowable limit.
This document discusses voltage and reactive power control methods in power systems. It covers the need for reactive power to maintain voltage levels and deliver active power through transmission lines. Various reactive power compensation devices are described such as series and shunt capacitors/reactors, synchronous condensers, static VAR compensators, and static synchronous compensators. Common voltage and reactive power control methods include excitation control at generating stations, using tap changing transformers, and switching shunt reactors/capacitors depending on load levels.
The document discusses multi-terminal DC (MTDC) systems. MTDC systems are used when there are multiple terminals in an HVDC transmission system. There are two main types of MTDC configurations: series and parallel. Series MTDC connects terminals in series, while parallel MTDC allows terminals to adjust currents independently and keep voltages constant. Radial and mesh are examples of parallel MTDC network topologies. MTDC systems provide benefits over multiple two-terminal HVDC links such as reduced costs and losses as well as increased transmission capacity and flexibility.
Power Quality is a combination of Voltage profile, Frequency profile, Harmonics contain and reliability of power supply.
The Power Quality is defined as the degree to which the power supply approaches the ideal case of stable, uninterrupted, zero distortion and disturbance free supply.
Overhead line insulators are used to electrically isolate power line conductors from each other and supporting structures. They protect transmission lines from over-voltages caused by lightning and switching. The most common insulator materials are porcelain and glass. Pin insulators are used for voltages up to 33kV, while suspension insulators are preferred for higher voltages as they can be scaled more easily. Proper insulator selection and arrangement is needed to achieve uniform voltage distribution across the insulator string. Sag in overhead lines must be properly calculated to limit conductor tension within safe levels while minimizing material usage and clearance heights.
This document discusses distributed generation technologies and their integration with electric utility systems. It describes various DG technologies like reciprocating engines, combustion turbines, fuel cells, wind and solar. It also discusses power quality issues associated with DG integration like voltage regulation, harmonics and islanding. The document outlines operating conflicts that can arise with DG like interference with utility fault clearing, reclosing and voltage regulation equipment. Finally, it covers DG placement on low-voltage distribution networks using spot networks and network protector relays.
Power system security refers to the probability that a power system will remain stable and within acceptable operating limits given potential disturbances or contingencies. There are three main operating states: preventive, emergency, and restorative. In the preventive state, the system operates normally and can withstand credible contingencies. The emergency state occurs when limits are violated, and the goal is to relieve stress. In the restorative state, parts of the system have lost power and the goal is to restore the system to normal. Security assessment involves system monitoring, contingency analysis to evaluate risks, and preventive and corrective actions. On-line security assessment continuously monitors the system using real-time measurements and updates assessments as conditions change.
This document provides information about flexible AC transmission systems (FACTS) including opportunities for FACTS, types of FACTS controllers, and their relative importance. It discusses how FACTS controllers can control parameters like line impedance, phase angle, and voltage injection to regulate power flow. The key types of FACTS controllers are series, shunt, and combined series-series or series-shunt configurations. Series controllers directly impact current and power flow, while shunt controllers control voltage. Combined controllers allow coordinated control and real power transfer between elements.
This document discusses future trends in electrical distribution system planning. It predicts that distribution planning will rely heavily on computer tools to optimize network design based on multiple criteria. Load management will also impact distribution by altering load patterns, requiring systems to be designed differently. New automated tools like network editors are expected to enable trial network designs that can be simulated to ensure performance and accommodate load growth. The central role of databases and management systems in supporting these new planning tools is also highlighted.
EC8553 Discrete time signal processing ssuser2797e4
This document contains a 10 question, multiple choice exam on discrete time signal processing. It covers topics like the discrete Fourier transform (DFT), finite word length effects, fixed point vs floating point representation, and FIR filter design. Specifically, it includes questions that calculate the 4 point DFT of a sequence, define twiddle factors, compare DIT and DIF FFT algorithms, and discuss stability and causality of systems.
The document discusses various objectives and applications of static shunt compensation on transmission lines. Shunt compensation can increase steady-state transmittable power, control voltage profiles, minimize line overvoltage under light loads using shunt reactors, and maintain voltage levels under heavy loads using shunt capacitors. Midpoint shunt compensation significantly increases transmitted power and is best located at the midpoint where voltage sag is maximum. End of line shunt compensation effectively increases voltage stability limits and regulates terminal voltages to prevent voltage instability. Shunt compensation can also improve transient stability and damp power oscillations on transmission lines.
Instantaneous Reactive Power Theory And Its Applicationsarunj89
Instantaneous Reactive Power Theory and its Applications to Active Power Filtering
The document discusses instantaneous reactive power (P-Q) theory, which was introduced by Hirofumi Akagi in 1983. P-Q theory defines instantaneous real and imaginary powers in the time domain, allowing it to be applied to non-sinusoidal systems. It has been widely used for harmonic compensation in active power filters. The document outlines the mathematical basis of P-Q theory, including Clarke transformations, definitions of instantaneous real and imaginary powers, and applications for compensating nonlinear loads. It also discusses developments and applications of P-Q theory, including its use in simulation and compensation of harmonic currents.
NEW STATCOM CONTROL SCHEME FOR POWER QUALITY IMPROVEMENT IN WIND FARM.sannuthi yaramapu
Now a days we are facing so many problems with power quality issues. So in order to mitigate these problems and to improve the power quality we are using new STATCOM control scheme in wind farm.
The document discusses implementing FACTS controllers on a 370km transmission line model to improve performance. It describes FACTS, different types of FACTS controllers including thyristor controlled series compensation (TCSC) and shunt compensation. Simulation results show that with FACTS compensation, the line can transfer more active power with better voltage regulation and power factor compared to the uncompensated line. FACTS increases the line's power transfer capability.
This document discusses smart grids and the role of advanced metering infrastructure in India. It notes that India has one of the weakest electrical grids in the world with high transmission losses. A smart grid uses communication and information technologies to better manage electricity distribution and demand. Advanced metering infrastructure is a key component, allowing two-way communication between utilities and customers to provide energy usage data and enable demand response programs. This can help improve grid reliability and efficiency while empowering consumers.
This document describes a project to improve power quality using a Unified Power Quality Conditioner (UPQC). The UPQC compensates for voltage disturbances and improves current quality using active power filters. It maintains the load voltage despite supply variations. The document outlines the objectives, introduces UPQC components like the shunt and series active power filters, and describes the multivariable controller and Simulink model. The UPQC provides advantages like reduced harmonics, improved waveform quality, and balanced power factor.
This document discusses static shunt compensation on transmission lines. Shunt compensation can increase steady-state transmittable power and control voltage profiles by using shunt reactors to minimize overvoltage under light loads and shunt capacitors to maintain voltage levels under heavy loads. Midpoint shunt compensation regulates voltage along line segments by exchanging only reactive power at the midpoint, significantly increasing transmittable power as the midpoint has the maximum voltage sag. End of line shunt compensation also provides voltage support to prevent instability.
This document discusses the need for transmission interconnections and opportunities provided by FACTS (Flexible AC Transmission Systems) technology. It notes that India has generation surpluses in some grids but deficits in others, and interconnections allow sharing of power to reduce costs. FACTS devices can control power flows and enhance line capacity, enabling more economic energy transfers. They offer advantages over mechanical switching like reduced wear and ability to damp oscillations. FACTS technology opens opportunities to better utilize transmission assets by overcoming thermal, dielectric and stability limitations on line loadings.
Application of Capacitors to Distribution System and Voltage RegulationAmeen San
Application of Capacitors to
Distribution System and Voltage
Regulation
POWER FACTOR IMPROVEMENT,
System Harmonics
Voltage Regulation
Methods of Voltage Control
In microgrid, if fault occurs or any other contingency happens, then the problems would be created which are related to power flow, also there are various protection schemes are used for minimize or eliminate these problems.
Voltage control is used for reactive power balance and P-f control is used for active power control.
Various protection schemes such as, over current protection, differential protection scheme, zoning of network in adaptive protection scheme are used in microgrid system .
The document describes a Simulink model that was created to improve total harmonic distortion (THD) using a shunt active power filter. The model simulates a power system with a non-linear load connected to an ideal grid voltage. The shunt active power filter is connected 0.1 seconds after simulation start and works to compensate for harmonics by producing currents equal in magnitude but opposite in phase to the load harmonics. Simulation results show the THD is reduced from around 30.9% on the load side to 2.79% on the source side once the active filter is connected, below the maximum allowable limit.
This document discusses voltage and reactive power control methods in power systems. It covers the need for reactive power to maintain voltage levels and deliver active power through transmission lines. Various reactive power compensation devices are described such as series and shunt capacitors/reactors, synchronous condensers, static VAR compensators, and static synchronous compensators. Common voltage and reactive power control methods include excitation control at generating stations, using tap changing transformers, and switching shunt reactors/capacitors depending on load levels.
The document discusses multi-terminal DC (MTDC) systems. MTDC systems are used when there are multiple terminals in an HVDC transmission system. There are two main types of MTDC configurations: series and parallel. Series MTDC connects terminals in series, while parallel MTDC allows terminals to adjust currents independently and keep voltages constant. Radial and mesh are examples of parallel MTDC network topologies. MTDC systems provide benefits over multiple two-terminal HVDC links such as reduced costs and losses as well as increased transmission capacity and flexibility.
Power Quality is a combination of Voltage profile, Frequency profile, Harmonics contain and reliability of power supply.
The Power Quality is defined as the degree to which the power supply approaches the ideal case of stable, uninterrupted, zero distortion and disturbance free supply.
Overhead line insulators are used to electrically isolate power line conductors from each other and supporting structures. They protect transmission lines from over-voltages caused by lightning and switching. The most common insulator materials are porcelain and glass. Pin insulators are used for voltages up to 33kV, while suspension insulators are preferred for higher voltages as they can be scaled more easily. Proper insulator selection and arrangement is needed to achieve uniform voltage distribution across the insulator string. Sag in overhead lines must be properly calculated to limit conductor tension within safe levels while minimizing material usage and clearance heights.
This document discusses distributed generation technologies and their integration with electric utility systems. It describes various DG technologies like reciprocating engines, combustion turbines, fuel cells, wind and solar. It also discusses power quality issues associated with DG integration like voltage regulation, harmonics and islanding. The document outlines operating conflicts that can arise with DG like interference with utility fault clearing, reclosing and voltage regulation equipment. Finally, it covers DG placement on low-voltage distribution networks using spot networks and network protector relays.
Power system security refers to the probability that a power system will remain stable and within acceptable operating limits given potential disturbances or contingencies. There are three main operating states: preventive, emergency, and restorative. In the preventive state, the system operates normally and can withstand credible contingencies. The emergency state occurs when limits are violated, and the goal is to relieve stress. In the restorative state, parts of the system have lost power and the goal is to restore the system to normal. Security assessment involves system monitoring, contingency analysis to evaluate risks, and preventive and corrective actions. On-line security assessment continuously monitors the system using real-time measurements and updates assessments as conditions change.
This document provides information about flexible AC transmission systems (FACTS) including opportunities for FACTS, types of FACTS controllers, and their relative importance. It discusses how FACTS controllers can control parameters like line impedance, phase angle, and voltage injection to regulate power flow. The key types of FACTS controllers are series, shunt, and combined series-series or series-shunt configurations. Series controllers directly impact current and power flow, while shunt controllers control voltage. Combined controllers allow coordinated control and real power transfer between elements.
This document discusses future trends in electrical distribution system planning. It predicts that distribution planning will rely heavily on computer tools to optimize network design based on multiple criteria. Load management will also impact distribution by altering load patterns, requiring systems to be designed differently. New automated tools like network editors are expected to enable trial network designs that can be simulated to ensure performance and accommodate load growth. The central role of databases and management systems in supporting these new planning tools is also highlighted.
EC8553 Discrete time signal processing ssuser2797e4
This document contains a 10 question, multiple choice exam on discrete time signal processing. It covers topics like the discrete Fourier transform (DFT), finite word length effects, fixed point vs floating point representation, and FIR filter design. Specifically, it includes questions that calculate the 4 point DFT of a sequence, define twiddle factors, compare DIT and DIF FFT algorithms, and discuss stability and causality of systems.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
This document provides an introduction to discrete-time signals and linear time-invariant (LTI) systems. It defines discrete-time signals as sequences represented at discrete time instants. Basic discrete-time sequences including the unit sample, unit step, and periodic sequences are described. Discrete-time systems are defined as transformations that map an input sequence to an output sequence. Linear and time-invariant systems are introduced. For LTI systems, the impulse response is defined and convolution is used to represent the output as a summation of the input multiplied by delayed versions of the impulse response. Key properties of LTI systems including superposition, scaling, time-invariance, and the commutative property of convolution are covered.
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
This document summarizes a presentation on computing divergences and distances between high-dimensional probability density functions (pdfs) represented using tensor formats. It discusses:
1) Motivating the problem using examples from stochastic PDEs and functional representations of uncertainties.
2) Computing Kullback-Leibler divergence and other divergences when pdfs are not directly available.
3) Representing probability characteristic functions and approximating pdfs using tensor decompositions like CP and TT formats.
4) Numerical examples computing Kullback-Leibler divergence and Hellinger distance between Gaussian and alpha-stable distributions using these tensor approximations.
This doctoral thesis models pulse propagation in optical fibers using finite-difference methods. It summarizes the numerical model, which solves the nonlinear Schrödinger equation describing pulse propagation using Crank-Nicholson and split-step Fourier methods. It tests the model's accuracy by comparing results to analytic solutions and a commercial simulation program. Effects like dispersion, loss, self-phase modulation and polarization mode dispersion are modeled. Additional models are presented for optical amplifiers and filters used in the fiber links.
Dsp 2018 foehu - lec 10 - multi-rate digital signal processingAmr E. Mohamed
This document discusses multi-rate digital signal processing and concepts related to sampling continuous-time signals. It begins by introducing discrete-time processing of continuous signals using an ideal continuous-to-discrete converter. It then covers the Nyquist sampling theorem and relationships between continuous and discrete Fourier transforms. It discusses ideal and practical reconstruction using zero-order hold and anti-imaging filters. Finally, it introduces the concepts of downsampling and upsampling in multi-rate digital signal processing systems.
DSP_FOEHU - Lec 08 - The Discrete Fourier TransformAmr E. Mohamed
The document discusses the Discrete Fourier Transform (DFT). It explains that while the discrete-time Fourier transform (DTFT) and z-transform are not numerically computable, the DFT avoids this issue. The DFT represents periodic sequences as a sum of complex exponentials with frequencies that are integer multiples of the fundamental frequency. It can be viewed as computing samples of the DTFT or z-transform at discrete frequency points, allowing numerical computation. The DFT provides a link between the time and frequency domain representations of a finite-length sequence.
The document discusses proportional (P), integral (I), and derivative (D) controllers which are commonly used in closed-loop control systems. It provides examples of how each controller type affects characteristics like rise time, overshoot, settling time, and steady-state error. A P controller reduces rise time but cannot eliminate steady-state error. An I controller eliminates steady-state error but can increase overshoot and settling time. A D controller reduces overshoot and improves transient response. The document recommends using PID control to achieve the best response by reducing rise time, overshoot, steady-state error, and improving stability.
The document discusses proportional (P), integral (I), and derivative (D) controllers which are commonly used in closed-loop control systems. It provides examples of how each controller type affects characteristics like rise time, overshoot, settling time, and steady-state error. A P controller reduces rise time but cannot eliminate steady-state error. An I controller eliminates steady-state error but can increase overshoot and settling time. A D controller reduces overshoot and improves transient response. The document recommends using PID control to achieve the best response by reducing rise time, overshoot, steady-state error, and improving stability.
Signals and Systems is an introduction to analog and digital signal processing, a topic that forms an integral part of engineering systems in many diverse areas, including seismic data processing, communications, speech processing, image processing, defense electronics, consumer electronics, and consumer products.
Continuous and Discrete Elementary signals,continuous and discrete unit step signals,Exponential and Ramp signals,continuous and discrete convolution time signal,Adding and subtracting two given signals,uniform random numbers between (0, 1).,random binary wave,random binary wave,robability density functions. Find mean and variance for the above
distributions
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
MIXED SPECTRA FOR STABLE SIGNALS FROM DISCRETE OBSERVATIONSsipij
This paper proposes a method to estimate the spectral density of a continuous-time stable alpha symmetric process from discrete observations of the process. Specifically, it considers when the spectral measurement is a mixture of a continuous component and discrete jumps. It samples the process at periodic times to create a periodogram, which is shown to be an asymptotically unbiased but inconsistent estimator. The periodogram is then smoothed using two spectral windows to account for the bandwidth of the spectral density, providing a consistent estimator of the spectral density at the jump points.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.
This document presents a seminar on a novel three-port converter (TPC) with high voltage gain for standalone renewable power systems. The TPC uses only three switches to control power flow between an input renewable energy source port, a bidirectional battery storage port, and a high-voltage load port. It can achieve a higher voltage gain for both low-voltage ports using a lower turns ratio and reasonable duty ratio. Finally, the document discusses implementing the TPC with a PV system to power a brushless DC motor drive and presenting simulation results using Matlab/Simulink software.
This document outlines the requirements for a major project consisting of a seminar, project work, and internship for undergraduate students. It states that in their final semester, students must register for an internship and work on a project and seminar. They must submit reports and presentations on their seminar, project, and internship work which will be evaluated by internal and external examiners. The seminar report is worth 50 marks, the project work is worth 200 marks total with 60 for internal evaluation and 140 for external, and the internship is worth 50 external marks based on a report and oral presentation.
This document discusses analog wattmeters and power factor meters. It provides information on:
1) Electrodynamometer type wattmeters which use a moving coil instrument to measure power in AC and DC circuits. The torque equation shows deflecting torque is proportional to power.
2) Power factor meters of the dynamometer and induction type which measure the power factor in single and three phase circuits.
3) Construction details, operating theory, torque equations, advantages and disadvantages of various analog power measurement instruments are covered. Numerical problems are also included.
1. The document discusses analog ammeters and voltmeters, including their classification, operating torques (deflecting, controlling, and damping), and types (PMMC, moving iron, electrostatic).
2. It also covers instrument construction, torque equations, range extension, temperature effects, errors and compensation, advantages and disadvantages.
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Week 2 Material.pdf
1. Power System Protection
Prof A K Pradhan
Department of Electrical Engineering
Indian Institute of Technology Kharagpur
Module 02: Phasor Estimation
Lecture 01 : Discrete Fourier Transform
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3. Phasor Estimation –
Discrete Fourier Transform(DFT)
• 1-cycle DFT
• Recursive DFT
• Half-cycle DFT
• Cosine Filter
Significance of phasors in relays- usage in most of the relays
sinusoid
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4. Discrete Fourier Transform(DFT)
Signal: 𝑣𝑣 𝑡𝑡 = 𝑉𝑉
𝑝𝑝 sin 𝜔𝜔𝑡𝑡 + 𝜃𝜃
𝑣𝑣𝑛𝑛 = 𝑉𝑉
𝑝𝑝 sin 𝜔𝜔𝑡𝑡𝑛𝑛 + 𝜃𝜃 where, 𝑡𝑡𝑛𝑛 = 𝑛𝑛∆𝑡𝑡; 𝑛𝑛 = 0,1,2, … … , ∆t =
time interval between sucessive samples
Data sampling: 8 samples/cycle
Sampling:
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5. Phasor estimation: 1-cycle DFT
Defining
𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 =
2
𝑁𝑁
�
𝑛𝑛=0
𝑁𝑁−1
[𝑣𝑣𝑛𝑛 cos(2𝜋𝜋
𝑛𝑛
𝑁𝑁
)] and 𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 =
2
𝑁𝑁
�
𝑛𝑛=0
𝑁𝑁−1
[𝑣𝑣𝑛𝑛 sin(2𝜋𝜋
𝑛𝑛
𝑁𝑁
)]
Computed Phasor:
̇
𝑉𝑉 = 𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑗𝑗𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑉𝑉 ∠𝜃𝜃
Where 𝑉𝑉 = 𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
2
+ 𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
2
,𝜃𝜃 = −tan−1(
𝑉𝑉𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
)
𝑣𝑣𝑛𝑛 = 𝑉𝑉
𝑝𝑝 sin 𝜔𝜔𝑡𝑡𝑛𝑛 + 𝜃𝜃
Applying 1-cycle DFT ,
Voltage phasor, ̇
𝑉𝑉 =
2
𝑁𝑁
∑𝑛𝑛=0
𝑁𝑁−1
(𝑣𝑣𝑛𝑛𝑒𝑒−𝑗𝑗
2𝜋𝜋
𝑁𝑁
𝑛𝑛
) ; 0 ≤ 𝑛𝑛 ≤ 𝑁𝑁 − 1 Where, N=number of samples in a cycle
𝑣𝑣𝑛𝑛 = 𝑛𝑛𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑣𝑣(𝑡𝑡)
�
𝑛𝑛=0
𝑁𝑁−1
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6. Data Sample window
𝑣𝑣𝑛𝑛 = 109.53 sin 100𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25° (V) ,sampling rate of 0.4 kHz, N=8
Time(s)
0.1 41.47
0.1025 101.01
0.105 101.37
0.1075 42.36
0.11 -41.47
0.1125 -101.01
0.1150 -101.37
0.1175 -42.36
0.12 41.47
0.1225 101.01
Window1
Window2
𝑣𝑣𝑛𝑛(𝑉𝑉)
Window1
Window2
-Moving window
-with new sample
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9. Observations: 1-cycle DFT
With the arrival of a new sample , the window is updated and the new
phasor advances by an angle of 450 .
the magnitude of the estimated phasor is same in both windows
There is a phase difference of 450 for the estimated phasors between
window 1 and window 2 ??
∆t=0.02/8 corresponds to 3600/8 = 450
450 ̇
𝑉𝑉𝑊𝑊𝑊
̇
𝑉𝑉𝑊𝑊𝑊= 77.45∠ − 67.75° (V)
= 77.45∠ − 22.75° (V)
Window1
Window2
∆t
w
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10. Phasor estimation in the presence of harmonics (1-cycle DFT)
𝑣𝑣𝑛𝑛 = 109.53 sin 100𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25° +5.48 sin 200𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25° +16.43 sin 300𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25°
+10.95 sin 500𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25° (V)
Sampling rate same, 0.4 kHz, N=8
Time(s) 𝑣𝑣𝑛𝑛(V)
0.1 53.92
0.1025 102.33
0.105 94.23
0.1075 48.20
0.11 -49.77
0.1125 -92.19
0.1150 -98.38
0.1175 -58.34
0.12 53.92
0.1225 102.33
Window
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14. 1-cycle DFT-phasors
0 0.01 0.02 0.03 0.04 0.05 0.06
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
time(s)
vabc(V)
window
Va∠θa
Vb∠θb
Vc∠θc
V1∠θ1
V2∠θ2
V0∠θ0
*****
Sequence
components
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15. Power System Protection
Prof A K Pradhan
Department of Electrical Engineering
Indian Institute of Technology Kharagpur
Module 02: Phasor Estimation
Lecture 07 : Recursive and Half Cycle DFT and Cosine Filter
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16. Phasor estimation techniques
Discrete Fourier Transform
Lecture 07
One cycle DFT, Recursive and Half Cycle DFT and Cosine Filter
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17. Recursive DFT:
𝑣𝑣𝑛𝑛 = 109.53 sin 100𝜋𝜋𝑡𝑡𝑛𝑛 + 𝜃𝜃 (V), 𝑡𝑡𝑛𝑛 = 𝑛𝑛∆𝑡𝑡, where ∆𝑡𝑡=0.0025 s
1-cycle DFT for window2 (𝑣𝑣1 through 𝑣𝑣8)
̇
𝑉𝑉2 =
2
𝑁𝑁
�
𝑛𝑛=0
𝑁𝑁−1
𝑣𝑣𝑛𝑛+1𝑒𝑒−𝑗𝑗
2𝜋𝜋𝑛𝑛
𝑁𝑁
Window1 Window2
Common Portion (𝑣𝑣1 – 𝑣𝑣7)
Outgoing Sample New sample
1-cycle DFT for window1 (𝑣𝑣0 through 𝑣𝑣7)
̇
𝑉𝑉1 =
2
𝑁𝑁
�
𝑛𝑛=0
𝑁𝑁−1
𝑣𝑣𝑛𝑛𝑒𝑒−𝑗𝑗
2𝜋𝜋𝑛𝑛
𝑁𝑁
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19. The phasor at the 𝑟𝑟 + 1 𝑡𝑡𝑡 instant can be written as
Earlier phasor New Sample Outgoing Sample
̇
𝑉𝑉𝑟𝑟+1 = [ ̇
𝑉𝑉
𝑟𝑟 +
2
𝑁𝑁
(𝑣𝑣N+r − 𝑣𝑣𝑟𝑟)]𝑒𝑒𝑗𝑗
2𝜋𝜋
𝑁𝑁
New phasor
Recursive DFT:
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23. Example-Half-cycle DFT:
𝑣𝑣𝑛𝑛 = 109.53 sin 100𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25° (V), N=8
Window 1
Window 2
Time(s)
0.1 41.47
0.1025 101.01
0.105 101.37
0.1075 42.36
0.11 -41.47
0.1125 -101.01
0.1150 -101.37
0.1175 -42.36
0.12 41.47
0.1225 101.01
Window1
Window2
𝑣𝑣𝑛𝑛(𝑉𝑉)
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24. Half-cycle DFT for window 1 (0.1s to 0.1075 s, 4 points)
For window1, ̇
𝑉𝑉 =
2 2
8
82.94 − 𝑗𝑗202.74
Time(s) Voltage Sample (𝒗𝒗𝒏𝒏) 𝑐𝑐𝑐𝑐𝑐𝑐(2𝜋𝜋
𝑛𝑛
𝑁𝑁
) 𝑠𝑠𝑠𝑠𝑠𝑠(2𝜋𝜋
𝑛𝑛
𝑁𝑁
) 𝑣𝑣𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠(2𝜋𝜋
𝑛𝑛
𝑁𝑁
)
𝑣𝑣𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐(2𝜋𝜋
𝑛𝑛
𝑁𝑁
)
0.1
0.1025
0.105
0.1075
41.47
101.01
101.37
42.36
⁄
1 2 ⁄
1 2
⁄
−1 2 ⁄
1 2
0
0
1
1
41.47
71.42 71.42
-29.95 29.95
0
0
101.37
82.94 202.74
= 77.45∠ − 67.75° (V)
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25. Half-cycle DFT computation for window 2(0.1025s to 0.11 s, 4 points)
For window2, ̇
𝑉𝑉 =
2 2
8
202.01 − 𝑗𝑗84.72
Time(s) Voltage Sample (𝒗𝒗𝒏𝒏) 𝑐𝑐𝑐𝑐𝑐𝑐(2𝜋𝜋
𝑛𝑛
𝑁𝑁
) 𝑠𝑠𝑠𝑠𝑠𝑠(2𝜋𝜋
𝑛𝑛
𝑁𝑁
) 𝑣𝑣𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠(2𝜋𝜋
𝑛𝑛
𝑁𝑁
)
𝑣𝑣𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐(2𝜋𝜋
𝑛𝑛
𝑁𝑁
)
⁄
1 2 ⁄
1 2
⁄
−1 2 ⁄
1 2
0
0
1
1
= 77.45∠ − 22.75° (V)
0.1025
0.105
0.1075
0.11
101.01
101.37
42.36
-41.47 29.32
101.01 0
0 42.36
71.68
-29.32
71.68
202.01 84.72
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26. Half-cycle DFT: Remarks
phasor is obtained with less number of samples.
Calculation is less as compared to one cycle DFT
During fault-it can provide phasor quickly
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27. Cosine Filter for Phasor Calculation
̇
𝑉𝑉 =
2
𝑁𝑁
�
𝑛𝑛=0
𝑁𝑁−1
𝑣𝑣𝑛𝑛𝑒𝑒−𝑗𝑗
2𝜋𝜋
𝑁𝑁 𝑛𝑛
; 0 ≤ 𝑛𝑛 ≤ 𝑁𝑁 − 1
̇
𝑉𝑉 =𝑉𝑉
𝑐𝑐 − 𝑗𝑗𝑉𝑉
𝑠𝑠
𝑉𝑉
𝑠𝑠 =
2
𝑁𝑁
�
𝑛𝑛=0
𝑁𝑁−1
�𝑣𝑣𝑛𝑛 sin 2𝜋𝜋
𝑛𝑛
𝑁𝑁
)
R𝑒𝑒𝑒𝑒𝑒𝑒 ̇
𝑉𝑉 = 𝑉𝑉
𝑐𝑐 I𝑚𝑚𝑚𝑚𝑚𝑚 ̇
𝑉𝑉 = −𝑉𝑉
𝑠𝑠
and
𝑉𝑉
𝑐𝑐 =
2
𝑁𝑁
�
𝑛𝑛=0
𝑁𝑁−1
�𝑣𝑣𝑛𝑛 cos 2𝜋𝜋
𝑛𝑛
𝑁𝑁
)
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34. Observation
Window No. Vcr Vsr Phasor
Window 1 165.88 405.48
Window 2 404.02 169.44
Window 3 405.48 -165.88
Window 4 169.44 -404.02
77.45∠ − 67.75° (V)
77.45∠ − 22.75° (V)
77.45∠22.25° (V)
77.45∠67.25° (V)
𝑉𝑉
𝑠𝑠𝑠𝑠 = −𝑉𝑉
𝑐𝑐 𝑟𝑟−
𝑁𝑁
4
̇
𝑉𝑉
𝑟𝑟 =𝑉𝑉
𝑐𝑐𝑟𝑟 − 𝑗𝑗𝑉𝑉
𝑠𝑠𝑠𝑠 =𝑉𝑉
𝑐𝑐𝑟𝑟+𝑗𝑗𝑉𝑉𝑐𝑐 𝑟𝑟−
𝑁𝑁
4
For the case with N=8,
̇
𝑉𝑉𝑟𝑟=𝑉𝑉
𝑐𝑐𝑐𝑐+𝑗𝑗𝑗𝑗𝑐𝑐 𝑟𝑟−2
Example
For window4
̇
𝑉𝑉4 =
2
8
169.44 + 𝑗𝑗404.02
= 77.45∠67.25° (V)
*****
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35. Power System Protection
Prof A K Pradhan
Department of Electrical Engineering
Indian Institute of Technology Kharagpur
Module 02: Phasor Estimation
Lecture 08 : Least Square Technique
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36. Phasor estimation techniques
• Least Square Estimation technique
• Application to Phasor estimation
Lecture 08 Least Square Technique
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37. Least Square Estimation
Consider a set of measurements that satisfies
𝑎𝑎 + 𝑏𝑏𝑡𝑡0 = 𝑚𝑚0
𝑎𝑎 + 𝑏𝑏𝑡𝑡𝑛𝑛−1 = 𝑚𝑚𝑛𝑛−1
𝑎𝑎 + 𝑏𝑏𝑏𝑏 = 𝑚𝑚
m is the measurement set, ‘t’ the associated time index
a and b are unknown system parameters to be obtained
.
.
Where,
For ‘a set of n’ number of measurements, taken at a regular interval
.
.
.
.
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑚𝑚0, 𝑚𝑚1 … . 𝑚𝑚𝑛𝑛−1 are the measurements and 𝑡𝑡0, 𝑡𝑡1 … . 𝑡𝑡𝑛𝑛−1 are corresponding time index
𝑎𝑎 + 𝑏𝑏𝑡𝑡1 = 𝑚𝑚1
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38. If �
𝑎𝑎 and �
𝑏𝑏 are the estimated values
�
𝑎𝑎 + �
𝑏𝑏𝑡𝑡0 − 𝑚𝑚0 = 𝜖𝜖1
�
𝑎𝑎 + �
𝑏𝑏𝑡𝑡1 − 𝑚𝑚1 = 𝜖𝜖2
�
𝑎𝑎 + �
𝑏𝑏𝑡𝑡𝑛𝑛−1 − 𝑚𝑚𝑛𝑛−1 = 𝜖𝜖𝑛𝑛−1
for 𝑚𝑚0, 𝑚𝑚2 … . 𝑚𝑚𝑛𝑛−1 are the measurements.
𝜖𝜖0, 𝜖𝜖2 … . 𝜖𝜖𝑛𝑛−1 are the errors (residues)
Where,
Least Square Estimation
1 𝑡𝑡0
1 𝑡𝑡1
. .
. .
1 𝑡𝑡𝑛𝑛−1
�
𝑎𝑎
�
𝑏𝑏
−
𝑚𝑚0
𝑚𝑚1
.
.
𝑚𝑚𝑛𝑛−1
=
𝜖𝜖0
𝜖𝜖1
.
.
𝜖𝜖𝑛𝑛−1
[A] [X] – [m] = [𝜖𝜖]
unknown
[𝜖𝜖]= [A][X] – [m]
n ×1
2 ×1
n ×2
n ×1
measurement
.
.
.
.
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39. Least Square Estimation
[𝜖𝜖]= [A][X] – [m]
𝜖𝜖 𝑇𝑇 𝜖𝜖 = 𝐴𝐴𝑋𝑋 − 𝑚𝑚 𝑇𝑇 𝐴𝐴𝑋𝑋 − 𝑚𝑚
= [ 𝐴𝐴𝑋𝑋 𝑇𝑇
− 𝑚𝑚 𝑇𝑇
] [𝐴𝐴𝑋𝑋 − 𝑚𝑚]
= 𝐴𝐴𝑋𝑋 𝑇𝑇 𝐴𝐴𝐴𝐴 + 𝑚𝑚 𝑇𝑇[𝑚𝑚] − 𝐴𝐴𝑋𝑋 𝑇𝑇 𝑚𝑚 − 𝑚𝑚 𝑇𝑇 𝐴𝐴𝐴𝐴
𝑚𝑚 𝑇𝑇 𝐴𝐴𝐴𝐴 = 𝑚𝑚𝑇𝑇𝐴𝐴𝐴𝐴 𝑇𝑇
𝑚𝑚 : n ×1 𝑚𝑚 𝑇𝑇
: 1 × n
[A]: n × 2
[X]: 2 × 1
𝐴𝐴𝐴𝐴 : n × 1
𝑚𝑚 𝑇𝑇
𝐴𝐴𝐴𝐴 : 1 × 1
For the 1 × 1 matrix,
= 𝐴𝐴𝑋𝑋 𝑇𝑇[𝑚𝑚]
= [𝑋𝑋]𝑇𝑇
[𝐴𝐴]𝑇𝑇
[𝑚𝑚]
𝜖𝜖 𝑇𝑇[𝜖𝜖] = 𝑋𝑋𝑇𝑇𝐴𝐴𝑇𝑇[𝐴𝐴𝑋𝑋] + 𝑚𝑚𝑇𝑇 𝑚𝑚 − 2 𝑋𝑋 𝑇𝑇 𝐴𝐴 𝑇𝑇[𝑚𝑚]
Here
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40. Least Square Estimation
Differentiating the above equation w.r.t. [𝑋𝑋]
2[𝐴𝐴]𝑇𝑇
𝐴𝐴 [X] − 2 𝐴𝐴 𝑇𝑇
[𝑚𝑚] = 0
[𝐴𝐴]𝑇𝑇
𝐴𝐴 [𝑋𝑋] = 𝐴𝐴 𝑇𝑇
[𝑚𝑚]
𝑋𝑋 = 𝐴𝐴𝑇𝑇
𝐴𝐴 −1
[𝐴𝐴]𝑇𝑇
[𝑚𝑚]
𝜖𝜖 𝑇𝑇[𝜖𝜖] = 𝑋𝑋 𝑇𝑇 𝐴𝐴 𝑇𝑇[𝐴𝐴𝑋𝑋] + 𝑚𝑚𝑇𝑇 𝑚𝑚 − 2 𝑋𝑋 𝑇𝑇 𝐴𝐴 𝑇𝑇[𝑚𝑚]
unknown
when[𝐴𝐴] is a square matrix, the pseudo inverse becomes invese of [A]
𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎 + 𝑏𝑏𝑏𝑏 = 𝑚𝑚
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41. Least Square Estimation for Phasor estimation
where 𝑣𝑣𝑛𝑛 voltage sample at 𝑡𝑡𝑛𝑛 , 𝑉𝑉, 𝜃𝜃 are to be found out
𝑣𝑣𝑛𝑛 = 𝑉𝑉 sin 𝜔𝜔𝑡𝑡𝑛𝑛 + 𝜃𝜃
𝑣𝑣1 = 𝑉𝑉 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝑡𝑡1 + 𝜃𝜃
= 𝑉𝑉 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝑡𝑡1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝑉𝑉 sin 𝜃𝜃 cos 𝜔𝜔𝑡𝑡1
at 𝑡𝑡 = 𝑡𝑡1,
= 𝑉𝑉 cos 𝜃𝜃sin 𝜔𝜔𝑡𝑡1 + 𝑉𝑉 sin 𝜃𝜃 cos 𝜔𝜔𝑡𝑡1
= 𝑎𝑎11 𝑋𝑋1 + 𝑎𝑎12𝑋𝑋2 𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑎𝑎11 = sin 𝜔𝜔𝑡𝑡1 , 𝑎𝑎12 = cos(𝜔𝜔𝑡𝑡1)
at 𝑡𝑡 = 𝑡𝑡0,
= 𝑉𝑉 cos 𝜃𝜃sin 𝜔𝜔𝑡𝑡0 + 𝑉𝑉 sin 𝜃𝜃 cos 𝜔𝜔𝑡𝑡0
𝑣𝑣0 = 𝑉𝑉 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝑡𝑡0 + 𝜃𝜃
= 𝑉𝑉 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝑡𝑡0 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝑉𝑉 sin 𝜃𝜃 cos 𝜔𝜔𝑡𝑡0
two unknowns?
= 𝑎𝑎01𝑋𝑋1 + 𝑎𝑎02𝑋𝑋2
𝑋𝑋1 = 𝑉𝑉 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃, 𝑋𝑋2 = 𝑉𝑉 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
𝑎𝑎01 = sin 𝜔𝜔𝑡𝑡0 , 𝑎𝑎02 = cos(𝜔𝜔𝑡𝑡0)
Where,
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43. 𝑋𝑋1 = 𝑉𝑉 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 𝑋𝑋2 = 𝑉𝑉 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
𝑉𝑉 = 𝑋𝑋1
2
+ 𝑋𝑋2
2
Vrms =
𝑉𝑉
2
Number of unknowns = 2, we need at least 2 samples to obtain the phasor
or more can be included.
Say, with 1 cycle data in the window, for 50 Hz and sampling rate 0.4 kHz, 8 samples
Size of X = 2 x 1
Size of A = 8 x 2
Size of m = 8 x 1
𝜃𝜃 = tan−1
𝑋𝑋2
𝑋𝑋1
Least Square Estimation for Phasor
𝐴𝐴 𝑋𝑋 = [𝑚𝑚]
𝑋𝑋 = 𝐴𝐴𝑇𝑇𝐴𝐴 −1[𝐴𝐴]𝑇𝑇 [𝑚𝑚]
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44. samples are taken at a rate of 0.4 kHz,
V 𝑡𝑡 = 109.53 sin 100𝜋𝜋𝜋𝜋 + 22.25°
(V) , ∆𝑡𝑡 = 0.0025 𝑠𝑠
Least Square Estimation for Phasor
Example1
Time(s)
0.1 41.47
0.1025 101.01
0.105 101.37
0.1075 42.36
0.11 -41.47
0.1125 -101.01
0.1150 -101.37
0.1175 -42.36
0.12 41.47
0.1225 101.01
𝑣𝑣𝑛𝑛(𝑉𝑉)
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45. Time(s)
0.1 41.47
0.1025 101.01
0.105 101.37
0.1075 42.36
0.11 -41.47
0.1125 -101.01
0.1150 -101.37
0.1175 -42.36
0.12 41.47
0.1225 101.01
𝑣𝑣𝑛𝑛(𝑉𝑉)
t0
t1
Least Square Estimation for Phasor
Example1..
[𝑚𝑚] =
41.47
101.01
[𝑋𝑋] =
𝑉𝑉 cos 𝜃𝜃
𝑉𝑉 sin 𝜃𝜃
𝜔𝜔 = 2𝜋𝜋𝜋𝜋 = 2𝜋𝜋 50 = 100𝜋𝜋
[𝐴𝐴] =
sin 𝜔𝜔𝑡𝑡0 cos 𝜔𝜔𝑡𝑡0
sin 𝜔𝜔𝑡𝑡1 cos 𝜔𝜔𝑡𝑡1
=
0 1
1
2
1
2
assigning time for the calculation window, t0 = 0.0 s and t1 = 0.0025s
in [A]
For the corresponding samples as marked in the table
𝑋𝑋 = 𝐴𝐴𝑇𝑇𝐴𝐴 −1[𝐴𝐴]𝑇𝑇 [𝑚𝑚]
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47. 𝑉𝑉 = 𝑋𝑋1
2
+ 𝑋𝑋2
2 = 109 ⋅ 53 𝑉𝑉 , 𝑉𝑉(𝑟𝑟𝑟𝑟𝑟𝑟) = 77.45 (V)
𝜃𝜃 = tan−1
𝑋𝑋2
𝑋𝑋1
= 22.250
Estimated phasor is 77.45∠22.25° (V)
Least Square Estimation for Phasor
Example1..
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48. Time(s)
0.1 41.47
0.1025 101.01
0.105 101.37
0.1075 42.36
0.11 -41.47
0.1125 -101.01
0.1150 -101.37
0.1175 -42.36
0.12 41.47
0.1225 101.01
𝑣𝑣𝑛𝑛(𝑉𝑉)
t0
t1
[𝑚𝑚] =
101.01
101.37
[𝑋𝑋] =
𝑉𝑉 cos 𝜃𝜃
𝑉𝑉 sin 𝜃𝜃
𝜔𝜔 = 2𝜋𝜋𝜋𝜋 = 2𝜋𝜋 50 = 100𝜋𝜋
[𝐴𝐴] =
sin 𝜔𝜔𝑡𝑡0 cos 𝜔𝜔𝑡𝑡0
sin 𝜔𝜔𝑡𝑡1 cos 𝜔𝜔𝑡𝑡1
=
0 1
1
2
1
2
with t0 = 0.0 s and t1 = 0.0025s for matrix A
For the corresponding samples as marked in the table
Least Square Estimation for Phasor
Example2- Different window
𝑋𝑋 = 𝐴𝐴𝑇𝑇𝐴𝐴 −1[𝐴𝐴]𝑇𝑇 [𝑚𝑚]
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50. 𝑉𝑉 = 𝑋𝑋1
2
+ 𝑋𝑋2
2 = 109 ⋅ 53 (V)
𝑉𝑉(𝑟𝑟𝑟𝑟𝑟𝑟) = 77.45 (V)
𝜃𝜃 = tan−1
𝑋𝑋2
𝑋𝑋1
= 67.250
Estimated phasor is 77.45∠67.25° (V)
Least Square Estimation for Phasor
Example2..
in the second window we got 77.45∠67.25° (V).
There is a phase shift of 45°which is correct
for the 0.4 kHz sampling for 50 Hz signal N=8
In first window, we got phasor 77.45∠22.25° (V)
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51. Phasor estimation in the presence of harmonics (Least Square Estimation)
𝑣𝑣𝑛𝑛 = 109.53 sin 100𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25° +5.48 sin 200𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25° +16.43 sin 300𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25°
+10.95 sin 500𝜋𝜋𝑡𝑡𝑛𝑛 + 22.25° (V)
Sampling rate same, 0.4 kHz,50 Hz, N=8
Time(s) 𝑣𝑣𝑛𝑛(V)
0.1 53.92
0.1025 102.33
0.105 94.23
0.1075 48.20
0.11 -49.77
0.1125 -92.19
0.1150 -98.38
0.1175 -58.34
0.12 53.92
0.1225 102.33
Example 3
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52. Time(s) 𝑣𝑣𝑛𝑛(V)
0.1 53.92
0.1025 102.33
0.105 94.23
0.1075 48.20
0.11 -49.77
0.1125 -92.19
0.1150 -98.38
0.1175 -58.34
0.12 53.92
0.1225 102.33
t0
t1 [𝑚𝑚] =
53.92
102.33
[𝑋𝑋] =
𝑉𝑉 cos 𝜃𝜃
𝑉𝑉 sin 𝜃𝜃
𝜔𝜔 = 2𝜋𝜋𝜋𝜋 = 2𝜋𝜋 50 = 100𝜋𝜋
[𝐴𝐴] =
sin 𝜔𝜔𝑡𝑡0 cos 𝜔𝜔𝑡𝑡0
sin 𝜔𝜔𝑡𝑡1 cos 𝜔𝜔𝑡𝑡1
=
0 1
1
2
1
2
Only with 2 measurements:
Example 3:
𝑋𝑋 = 𝐴𝐴𝑇𝑇𝐴𝐴 −1[𝐴𝐴]𝑇𝑇 [𝑚𝑚]
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54. 𝑉𝑉 = 𝑋𝑋1
2
+ 𝑋𝑋2
2 = 105.6 (𝑉𝑉), 𝑉𝑉(𝑟𝑟𝑟𝑟𝑟𝑟) = 74.67 (V)
𝜃𝜃 = tan−1
𝑋𝑋2
𝑋𝑋1
= 30.70
Estimated phasor is 74.67∠30.7° (V)
correct phasor 77.45∠22.25° (V)
Example 3..
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55. Time(s) 𝑣𝑣𝑛𝑛(V)
0.1 53.92
0.1025 102.33
0.105 94.23
0.1075 48.20
0.11 -49.77
0.1125 -92.19
0.1150 -98.38
0.1175 -58.34
0.12 53.92
0.1225 102.33
[𝑚𝑚] =
53.92
102.33
94.23
48.20
−49.77
−92.19
−98.38
−58.34
[𝑋𝑋] =
𝑉𝑉 cos 𝜃𝜃
𝑉𝑉 sin 𝜃𝜃
𝜔𝜔 = 2𝜋𝜋𝜋𝜋 = 2𝜋𝜋 50 = 100𝜋𝜋
With 8 measurements (1-cycle window)
t0
t1
t2
t3
t4
t5
t6
t7
[𝐴𝐴] =
sin 𝜔𝜔𝑡𝑡0 cos 𝜔𝜔𝑡𝑡0
sin 𝜔𝜔𝑡𝑡1 cos 𝜔𝜔𝑡𝑡1
sin 𝜔𝜔𝑡𝑡2
sin 𝜔𝜔𝑡𝑡3
sin 𝜔𝜔𝑡𝑡4
sin 𝜔𝜔𝑡𝑡5
sin 𝜔𝜔𝑡𝑡6
sin 𝜔𝜔𝑡𝑡7
cos 𝜔𝜔𝑡𝑡2
cos 𝜔𝜔𝑡𝑡3
cos 𝜔𝜔𝑡𝑡4
cos 𝜔𝜔𝑡𝑡5
cos 𝜔𝜔𝑡𝑡6
cos 𝜔𝜔𝑡𝑡7
=
0 1
1
2
1
2
1
1
2
0
−
1
2
−1
−
1
2
−
0
1
2
−1
−
1
2
0
1
2
with t0 = 0.0 s and t1 = 0.0025s … t7 =0.175 s
for matrix A
Example 3:
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56. 𝐴𝐴𝑇𝑇
𝐴𝐴 −1
𝐴𝐴𝑇𝑇
=
=
101.37
41.47
0 0.1768 0.25 0.1768 0 −0.1768 −0.25 −0.1768
0.25 0.1768 0 −0.1768 −0.25 −0.1768 0 0.1768
𝑉𝑉 = 𝑋𝑋1
2
+ 𝑋𝑋2
2 = 109 ⋅ 53 (𝑉𝑉) 𝑉𝑉(𝑟𝑟𝑟𝑟𝑟𝑟) = 77.45 (V)
𝜃𝜃 = tan−1
𝑋𝑋2
𝑋𝑋1
= 22.250
Estimated phasor is 77.45∠22.25° (V)
This is correct the phasor.
𝑋𝑋 = 𝐴𝐴𝑇𝑇
𝐴𝐴 −1
[𝐴𝐴]𝑇𝑇
[𝑚𝑚]
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57. Estimation of harmonic component (using Least Square Estimation)
𝑣𝑣𝑛𝑛 = 𝑉𝑉1 sin 𝜔𝜔𝑡𝑡𝑛𝑛 + 𝜃𝜃1 +𝑉𝑉2 sin 2𝜔𝜔𝑡𝑡𝑛𝑛 + 𝜃𝜃2
𝑣𝑣𝑛𝑛 = 𝑉𝑉1 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝑡𝑡n 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃1 + 𝑉𝑉1 sin 𝜃𝜃1 cos 𝜔𝜔𝑡𝑡n +𝑉𝑉2 𝑠𝑠𝑠𝑠𝑠𝑠 2𝜔𝜔𝑡𝑡n 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃2 + 𝑉𝑉2 sin 𝜃𝜃2 cos 2𝜔𝜔𝑡𝑡n
Say we need 2nd harmonic component to be estimated with fundamental
A =
sin 𝜔𝜔𝑡𝑡0 cos 𝜔𝜔𝑡𝑡0 sin 2𝜔𝜔𝑡𝑡0 cos 2𝜔𝜔𝑡𝑡0
sin 𝜔𝜔𝑡𝑡1 cos 𝜔𝜔𝑡𝑡1 sin 2𝜔𝜔𝑡𝑡1 cos 2𝜔𝜔𝑡𝑡1
⋮ ⋮ ⋮ ⋮
⋮ ⋮ ⋮ ⋮
sin 𝜔𝜔𝑡𝑡n−1 cos 𝜔𝜔𝑡𝑡n−1 sin 2𝜔𝜔𝑡𝑡n−1 cos 2𝜔𝜔𝑡𝑡n−1
𝑚𝑚 =
𝑣𝑣0
𝑣𝑣1
⋮
⋮
𝑣𝑣n
𝑉𝑉1 = 𝑋𝑋1
2
+ 𝑋𝑋2
2 V1rms =
𝑉𝑉1
2
𝜃𝜃1 = tan−1
𝑋𝑋2
𝑋𝑋1
𝑋𝑋 =
𝑉𝑉1 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃1
𝑉𝑉1 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃1
𝑉𝑉2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃2
𝑉𝑉2 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃2
𝑉𝑉2 = 𝑋𝑋3
2
+ 𝑋𝑋4
2 𝜃𝜃2 = tan−1
𝑋𝑋4
𝑋𝑋3
V2rms =
𝑉𝑉2
2
𝑋𝑋2
𝑋𝑋1
𝑋𝑋3
𝑋𝑋4
2nd harmonic
𝑋𝑋 = 𝐴𝐴𝑇𝑇𝐴𝐴 −1[𝐴𝐴]𝑇𝑇 [𝑚𝑚]
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58. Remarks
Least square estimation- provides phasors like DFT
It can manage with less number of samples for pure sinusoid
But with harmonics– it is able to filter out with 1-cycle of data
—similar to 1-cycle DFT
- We can incorporate harmonics also and get the magnitude and phase.
To reduce computation- matrix- [A] is fixed for a given window size and
signal sampling rate—so also the 𝐴𝐴𝑇𝑇𝐴𝐴 −1[𝐴𝐴]𝑇𝑇
∗∗∗∗∗
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59. Power System Protection
Prof A K Pradhan
Electrical Engineering, IIT KHARAGPUR
Module 02: Phasor Estimation
Lecture 09 : Frequency Response of Phasor Estimation techniques
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60. Lecture 09: Frequency Response of Phasor Estimation techniques
• Frequency Response of different phasor estimation techniques
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61. Frequency Response of Filters
• The power system signal distortions- inrush, power electronics devices…
• It provides the response of a filter for different frequencies as input signal- which is important to
assess the performance, for obtaining fundamental component from a voltage/current signal which
may be distorted in the system
-it reveals the strength of the estimator
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63. • Frequency response of the filter can be obtained by substituting 𝑧𝑧 = 𝑒𝑒𝑗𝑗𝑗𝑗∆𝑡𝑡
Where ∆𝑡𝑡= sampling time interval and 𝜔𝜔=frequency of input signal.
Frequency Response: 1-cycle DFT
Cosine Filter
• For fundamental frequency input, 𝜔𝜔=2𝜋𝜋 x 50 = 100𝜋𝜋
𝐻𝐻𝑐𝑐 100𝜋𝜋 =
1
4
1.0
1
2
− 𝑗𝑗
1
2
+
1
2
0 − 𝑗𝑗𝑗 + 0.0 −
1
2
− 𝑗𝑗
1
2
−
1
2
−1 + 𝑗𝑗𝑗.0 −
1.0 −
1
2
+ 𝑗𝑗
1
2
−
1
2
0 + 𝑗𝑗𝑗 + 0.0
1
2
+ 𝑗𝑗
1
2
+
1
2
1 + 𝑗𝑗𝑗
=
1
4
4 ×
1
2
− 4 × 𝑗𝑗
1
2
=
1
2
− 𝑗𝑗
1
2
= 1∠ −
𝜋𝜋
4
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64. 𝐻𝐻𝑠𝑠 100𝜋𝜋 =
1
4
[0.0
1
2
− 𝑗𝑗
1
2
−
1
2
0 − 𝑗𝑗1 − 1.0 −
1
2
− 𝑗𝑗
1
2
−
1
2
−1 + 𝑗𝑗0.0
Frequency Response: 1-cycle DFT
Sine Filter
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65. Frequency Response: 1-cycle DFT
Power System Relaying Committee IEEE working group report, Understanding microprocessor based technology applied to relaying,
Feb2004
L. Wang, Frequency Response of Phasor based microprocessor relaying algorithms, IEEETransactions on Power Delivery, vol 14, no.1,
1999, page 98
For DC component, 𝜔𝜔=0
Hc (0) =
1
4
1.0 +
1
2
+ 0.0 −
1
2
− 1.0 −
1
2
+ 0.0 +
1
2
= 0
Hs (0) =
1
4
[0.0 −
1
2
− 1.0 −
1
2
− 0.0 +
1
2
+ 1.0 +
1
2
] = 0
For second harmonic, 𝜔𝜔=2𝜋𝜋 2 50 =200𝜋𝜋
Hc (200𝜋𝜋) =0
Hs (200𝜋𝜋) = 0
x x
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66. Frequency Response: 1-cycle DFT
Remarks
•All harmonics and DC component are removed
•Sine filter suppresses the high frequency components better than the Cosine filter
•Cosine filter suppresses the sub-harmonic components better than the Sine filters
•Sub-harmonics and inter-harmonics present in the signal will affect the phasor estimation accuracy
COS
SIN
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67. Frequency Response: ½ - cycle DFT
Sample No. Delays Sin(-∆θ)
0 3 1.0 0
1 2
2 1 0 -1
3 0
Using z-Transform
Hc(ω) =
1
4
[1.0𝑧𝑧3 +
1
2
𝑧𝑧2 + 0.0𝑧𝑧1 −
1
2
𝑧𝑧0]
Hs(ω) =
1
4
[0.0𝑧𝑧3 −
1
2
𝑧𝑧2 − 1.0𝑧𝑧1 −
1
2
𝑧𝑧0]
̇
𝑉𝑉 =
2
𝑁𝑁
2
�
𝑛𝑛=0
𝑁𝑁
2
−1
𝑣𝑣𝑛𝑛𝑒𝑒−𝑗𝑗
2𝜋𝜋
𝑁𝑁
𝑛𝑛
∆θ =
2π𝑛𝑛
𝑁𝑁
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70. Remarks-
• Frequency response shows – the rejection of filters to harmonics and DC–steady state
• One cycle DFT vs half cycle DFT
• Least square filter- with larger window
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71. Power System Protection
Prof A K Pradhan
Electrical Engineering, IIT KHARAGPUR
Module 02: Phasor Estimation
Lecture 10 : In the Presence of Decaying DC
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72. Lecture 10: In the Presence of Decaying DC
• Decaying DC issue in the fault signals
• Solution
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73. Presence of Decaying DC in current signal:
v(t)
R L
v t = V sin ωt + θ
i t = 𝐼𝐼𝑚𝑚[sin ωt + θ − θz − sin θ − θz e−
t
τ]
Here, θz = tan−1(
X
R
) =tan−1(
ωL
R
)
θ − θz = nπ ; zero transient, n=0,1,2,3..
θ − θz =
nπ
2
; maximum transient, n=1,3,5..
Thus, for different faults, fault inceptions, the relay will see different amount of decaying DC
The decaying DC will result in larger magnitude of phasor, leading to incorrect relay
decision
i t
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74. Solution: Elimination of Decaying DC using Mimic Filter :
i(t)
R’ L’
Let decaying dc, i t = e−
t
τ
Vo s = sL′
+ R′
I s
ℒ−1 Vo(s) = ℒ−1[
sL′+R′
s+
1
τ
]
= L′
ℒ−1
s+
1
𝜏𝜏
s+
1
τ
= L′
u(t)
𝑤𝑤𝑤𝑤𝑤𝑤𝑤, 𝜏𝜏 =
L′
R′
Vo(t)
Mimic impedance where u(t)– unit impulse
This implies, decaying DC in the output has vanished
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75. Relay current with Mimic Filter
• Decaying DC is filtered out.
• This introduces phase lag, that has to be compensated for correct phasor estimation.
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76. Elimination of Decaying DC using Digital Mimic Filter :
• With a mimic circuit consisting of R-L in impedance form K(1+sτ), then the exponentially decaying component at the output will
vanish, provided its time constant is equal to τ.
• The differentiator circuit, as with s-term, can be emulated by digital FIR filter: (1-z-1)
• The impedance can be represented as: K[(1+ τ) - τ z-1]
• K has to be set in such a way that, at rated frequency (50/60 Hz), the filter gain will be 1.
• The corresponding gain
f= 50/60 Hz
Gain(f) = |K[(1+ τ)- τ𝑒𝑒−𝑗𝑗𝑗𝑗∆𝑡𝑡
]| = 1
• Solving this equation for K, we obtain 𝐾𝐾2
=
1
1 + τ − τ cos 2𝜋𝜋/𝑁𝑁 2 + τ sin 2𝜋𝜋/𝑁𝑁 2
• Thus using mimic filter, the current sample at pth instant can be obtained as,
𝑖𝑖′ 𝑝𝑝 = 𝐾𝐾 1 + τ ∗ 𝑖𝑖 𝑝𝑝 − τ ∗ 𝑖𝑖(𝑝𝑝 − 1)
N = number of samples per cycle
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77. Fault current response using Digital Mimic Filter with phase shift compensation:
𝜙𝜙𝑠𝑠𝑠 = tan−1
𝜏𝜏 sin ⁄
2𝜋𝜋 𝑁𝑁
1 + 𝜏𝜏 − 𝜏𝜏 cos ⁄
2𝜋𝜋 𝑁𝑁
Phase shift with mimic filter,
𝜏𝜏 = decaying time constant
N = number of samples per cycle
G. Benmouyal, "Removal of DC-offset in current waveforms using digital mimic filtering," IEEE Trans. Power Del., vol. 10, no. 2, pp. 621-630, April
1995.
Gain(f) = |K[(1+ τ)- τ𝑒𝑒−𝑗𝑗𝑗𝑗∆𝑡𝑡
]| = 1
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78. Least Square Estimation in Presence of Decaying DC
𝐼𝐼, 𝜃𝜃, 𝑘𝑘0, τ are unknowns
𝑖𝑖𝑛𝑛 = 𝐼𝐼 sin 𝜔𝜔𝑡𝑡𝑛𝑛 + 𝜃𝜃 + 𝑘𝑘0𝑒𝑒−
𝑡𝑡𝑛𝑛
𝜏𝜏
𝑋𝑋1 = 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃, 𝑋𝑋2 = 𝐼𝐼 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃,
𝑎𝑎n1 = sin(𝜔𝜔𝑡𝑡n), 𝑎𝑎n2 = cos(𝜔𝜔𝑡𝑡n),
From Taylor series expansion
𝑘𝑘0𝑒𝑒−
𝑡𝑡𝑛𝑛
𝜏𝜏 = 𝑘𝑘0 − 𝑘𝑘0
𝑡𝑡𝑛𝑛
𝜏𝜏
+ 𝑘𝑘0
𝑡𝑡𝑛𝑛
2
2!𝜏𝜏2 − … … … …
Neglecting higher order terms,
𝑘𝑘0𝑒𝑒−
𝑡𝑡𝑛𝑛
𝜏𝜏 = 𝑘𝑘0 − 𝑘𝑘0
𝑡𝑡𝑛𝑛
𝜏𝜏
Thus, 𝑖𝑖𝑛𝑛 = 𝐼𝐼 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝑡𝑡n 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝐼𝐼 sin 𝜃𝜃 cos 𝜔𝜔𝑡𝑡n + 𝑘𝑘0 − 𝑘𝑘0
𝑡𝑡𝑛𝑛
𝜏𝜏
𝑋𝑋3 = 𝑘𝑘0, 𝑋𝑋4 =
𝑘𝑘0
𝜏𝜏
𝑎𝑎n3 = 1, 𝑎𝑎n4 = −𝑡𝑡n
𝑡𝑡𝑛𝑛 = 𝑛𝑛Δ𝑡𝑡; 0 ≤ 𝑛𝑛 ≤ 𝑁𝑁 − 1
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑛𝑛 is current sample at 𝑡𝑡𝑛𝑛
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79. Least Square Estimation in Presence of Decaying DC
A =
sin 𝜔𝜔𝑡𝑡0 cos 𝜔𝜔𝑡𝑡0 1 −𝑡𝑡0
sin 𝜔𝜔𝑡𝑡1 cos 𝜔𝜔𝑡𝑡1 1 −𝑡𝑡1
⋮ ⋮ ⋮ ⋮
⋮ ⋮ ⋮ ⋮
sin 𝜔𝜔𝑡𝑡n cos 𝜔𝜔𝑡𝑡n 1 −𝑡𝑡n
𝑋𝑋 =
𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃
𝐼𝐼 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
𝑘𝑘0
𝑘𝑘0
𝜏𝜏
𝑚𝑚 =
𝑖𝑖0
𝑖𝑖1
⋮
⋮
𝑖𝑖n
𝑋𝑋 = 𝐴𝐴𝑇𝑇𝐴𝐴 −1𝐴𝐴𝑇𝑇𝑚𝑚
𝐼𝐼 = 𝑋𝑋1
2
+ 𝑋𝑋2
2
Note, here I(rms) =
𝐼𝐼
2
𝜃𝜃 = tan−1
𝑋𝑋2
𝑋𝑋1
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80. Least Square Estimation in the Presence of Decaying DC: Example
𝑖𝑖𝑛𝑛 = 100 sin 100𝜋𝜋𝑡𝑡𝑛𝑛 + 30° + 250 𝑒𝑒−
𝑡𝑡𝑛𝑛
40 ) (A), 𝑡𝑡𝑛𝑛 = 𝑛𝑛∆𝑡𝑡, where ∆𝑡𝑡=0.0025 s
𝑚𝑚 =
299.38
345.95
335.95
275.21
A =
sin 𝜔𝜔𝑡𝑡0 cos 𝜔𝜔𝑡𝑡0 1 −𝑡𝑡0
sin 𝜔𝜔𝑡𝑡1 cos 𝜔𝜔𝑡𝑡1 1 −𝑡𝑡1
sin 𝜔𝜔𝑡𝑡2 cos 𝜔𝜔𝑡𝑡2 1 −𝑡𝑡2
sin 𝜔𝜔𝑡𝑡3 cos 𝜔𝜔𝑡𝑡3 1 −𝑡𝑡3
=
0 1 1 0
1
2
1
2
1 −
1
8
1 0 1 −
1
4
1
2
−
1
2
1 −
3
8
Using first four sample points (half cycle),
𝑋𝑋 =
𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃
𝐼𝐼 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃
𝑘𝑘0
𝑘𝑘0
𝜏𝜏
From Least Square technique, 𝑋𝑋 = 𝐴𝐴𝑇𝑇𝐴𝐴 −1𝑚𝑚
= 100
𝑋𝑋 =
86.60
50.00
249.38
0.12
I = 𝑋𝑋1
2
+ 𝑋𝑋2
2
𝜃𝜃 = tan−1
𝑋𝑋2
𝑋𝑋1
= 300
(A)
= 86.62
+ 502
Estimated current
𝐼𝐼(𝑟𝑟𝑟𝑟𝑟𝑟) = 70.7(A)
Estimated current phasor = 70.7∠30° (A) This is the correct phasor.
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81. Remarks
• Decaying DC will affect phasor values unless being filtered out-
affects relay performance like- distance relay underreach
• Mitigation-
Mimic filter approach with DFT
• Least square Technique– in the modelling we can incorporate
*******
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