School of Engineering (SOE) University of St Thomas (UST)
Transmission Lines and Power Flow Analysis
Dr. Greg Mowry Annie Sebastian Marian Mohamed
1 School of Engineering
The UST Mission
Inspired by Catholic intellectual tradition, the University of St. Thomas educates students to be morally responsible leaders who think critically, act wisely, and work skillfully to advance the common good. http://www.stthomas.edu/mission/
Since civilization depends on power, power is for the ‘common good’!! 2 School of Engineering
Outline
I. Background material
II. Transmission Lines (TLs)
III. Power Flow Analysis (PFA)
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I. Background Material
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Networks & Power Systems
In a network (power system) there are 6 basic electrical quantities of interest:
1. Current 푖 푡 = 푑푞/푑푡
2. Voltage 푣 푡 = 푑휑/푑푡 (FL)
푑푤 푑퐸 3. Power 푝 푡 = 푣 푡 푖 푡 = = 푑푡 푑푡 5 School of Engineering
Networks & Power Systems
6 basic electrical quantities continued:
푡 푡 4. Energy (work) w t = −∞ 푝 휏 푑휏 = −∞ 푣 휏 푖 휏 푑휏
푡 5. Charge (q or Q) 푞 푡 = −∞ 푖 휏 푑휏
푡 6. Flux 휑 푡 = −∞ 푣 휏 푑휏
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Networks & Power Systems
A few comments:
Voltage (potential) has potential-energy characteristics and therefore needs a reference to be meaningful; e.g. ground 0 volts
I will assume that pretty much everything we discuss & analyze today is linear until we get to PFA
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Networks & Power Systems
Linear Time Invariant (LTI) systems:
Linearity:
If 푦1 = 푓 푥1 & 푦2 = 푓(푥2)
Then 훼 푦1 + 훽 푦2 = 푓(훼 푥1 + 훽 푥2)
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Networks & Power Systems
Implications of LTI systems:
Scalability: if 푦 = 푓(푥), then 훼푦 = 푓(훼푥)
Superposition & scalability of inputs holds
Frequency invariance: 휔푖푛푝푢푡 = 휔표푢푡푝푢푡
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ACSS and LTI systems:
Suppose: 푣푖푛 푡 = 푉푖푛 cos 휔푡 + 휃
Since input = output, the only thing a linear system can do to an input is:
. Change the input amplitude: Vin Vout
. Change the input phase: in out 10 School of Engineering
Networks & Power Systems
ACSS and LTI systems:
These LTI characteristics along with Euler’s theorem
푒푗휔푡 = cos 휔푡 + 푗 sin 휔푡 ,
allow 푣푖푛 푡 = 푉푖 cos 휔푡 + 휃 to be represented as
푗휃 푣푖푛 푡 = 푉푖 cos 휔푡 + 휃 푉푖푒 = 푉푖 ∠휃 11 School of Engineering
Networks & Power Systems
In honor of Star Trek,
푗휃 푉푖푒 = 푉푖 ∠휃
is called a ‘Phasor’
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푗휃 푉푖푒 = 푉푖 ∠휃 13 School of Engineering
Networks & Power Systems
ACSS and LTI systems:
Since time does not explicitly appear in a phasor, an LTI system in ACSS can be treated like a DC system on steroids with j
ACSS also requires impedances; i.e. AC resistance
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Networks & Power Systems
Comments: Little, if anything, in ACSS analysis is gained by thinking of j = −1 (that is algebra talk)
Rather, since Euler Theorem looks similar to a 2D vector 푒푗휔푡 = cos 휔푡 + 푗 sin 휔푡
푢 = 푎 푥 + 푏 푦 15 School of Engineering
푒푗90° = cos 90° + 푗 sin 90° = 푗
Hence in ACSS analysis, think of j as a 90 rotation
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Operators v(t) V V
dv dt jV
V vdt j
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Ohm’s Approximation (Law)
Summary of voltage-current relationship Element Time domain Frequency domain R v Ri V RI L di v L V jLI dt C dv I i C V dt jC
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Impedance & Admittance
Impedances and admittances of passive elements Element Impedance Admittance R 1 Y Z R R 1 L Z jL Y jL 1 C Z Y jC jC 18 School of Engineering
The Impedance Triangle
Z = R + j X 18 School of Engineering
The Power Triangle (ind. reactance)
S = P + j Q 18 School of Engineering
Triangles
Comments:
Power Triangle = Impedance Triangle
The cos() defines the Power Factor (pf)
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Reminder of Useful Network Theorems
KVL – Energy conservation 푣퐿표표푝 = 0
KCL – Charge conservation 푖푁표푑푒 표푟 푏푢푠 = 0
Ohm’s Approximation (Law)
Passive sign Convention (PSC)
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Passive Sign Convection – Power System
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PSC Source Load or Circuit Element
PSource = vi PLoad = vi 25 PSource = PLoad School of Engineering
Reminder of Useful Network Theorems
Pointing Theorem 푆 = 푉 퐼∗
Thevenin Equivalent – Any linear circuit may be represented by a voltage source and series Thevenin impedance
Norton Equivalent – Any linear circuit my be represented by a current source and a shunt Thevenin impedance 26 푉푇퐻 = 퐼푁 푍푇퐻 School of Engineering (SOE) University of St Thomas (UST)
Transmission Lines and Power Flow Analysis
Dr. Greg Mowry Annie Sebastian Marian Mohamed
1 School of Engineering
II. Transmission Lines
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Outline
General Aspects of TLs
TL Parameters: R, L, C, G
2-Port Analysis (a brief interlude)
Maxwell’s Equations & the Telegraph equations
Solutions to the TL wave equations 3 School of Engineering
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Transmission Lines (TLs)
A TL is a major component of an electrical power system.
The function of a TL is to transport power from sources to loads with minimal loss.
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Transmission Lines
In an AC power system, a TL will operate at some frequency f; e.g. 60 Hz.
Consequently, 휆 = 푐/푓
The characteristics of a TL manifest
themselves when 흀 ~ 푳 of the system. 7 School of Engineering
USA example
The USA is ~ 2700 mi. wide & 1600 mi. from north-to-south (4.35e6 x 2.57e6 m).
3푒8 For 60 Hz, 휆 = = 5푒6 푚 60
Thus 흀ퟔퟎ 푯풛 ~ 푳푼푺푨 and consequently the continental USA transmission system will exhibit TL characteristics 8 School of Engineering
Overhead TL Components
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Overhead ACSR TL Cables
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Some Types of Overhead ACSR TL Cables
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TL Parameters
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Model of an Infinitesimal TL Section
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General TL Parameters a. Series Resistance – accounts for Ohmic (I2R losses) b. Series Impedance – accounts for series voltage drops Resistive Inductive reactance c. Shunt Capacitance – accounts for Line-Charging Currents d. Shunt Conductance – accounts for V2G losses due to leakage currents between conductors or between conductors and ground. 14 School of Engineering
Power TL Parameters
Series Resistance: related to the physical structure of the TL conductor over some temperature range.
Series Inductance & Shunt Capacitance: produced by magnetic and electric fields around the conductor and affected by their geometrical arrangement.
Shunt Conductance: typically very small so neglected. 15 School of Engineering
TL Design Considerations
Design Considerations Responsibilities
1. Dictates the size, type and number of bundle conductors per phase. 2. Responsible for number of insulator discs, vertical or v-shaped arrangement, Electrical Factors phase to phase clearance and phase to tower clearance to be used 3. Number, Type and location of shield wires to intercept lightning strokes. 4. Conductor Spacing's, Types and sizes
Mechanical Factors Focuses on Strength of the conductors, insulator strings and support structures
Environmental Factors Include Land usage, and visual impact
Economic Factors Technical Design criteria at lowest overall cost
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Transmission Line Components
Components Made of Types ACSR - Aluminum Conductor Steel Reinforced AAC - All Aluminum Conductor AAAC - All Aluminum Alloy Conductor ACAR - Aluminum Conductor Alloy Reinforced Aluminum replaced Conductors Alumoweld - Aluminum clad Steel Conductor copper ACSS - Aluminum Conductor Steel Supported GTZTACSR - Gap-Type ZT Aluminum Conductor ACFR - Aluminum Conductor Carbon Reinforced ACCR - Aluminum Conductor Composite Reinforced Pin Type Insulator Porcelain, Toughened Insulators Suspension Type Insulator glass and polymer Strain Type Insulator Steel, Alumoweld Shield wires Smaller Cross section compared to 3 phase conductors and ACSR Wooden Poles Lattice steel Tower, Reinforced Concrete Poles Support Structures Wood Frame Steel Poles Lattice Steel towers. 17 School of Engineering
ACSR Conductor Characteristic
Overall Dia DC Resistance Current Capacity Name (mm) (ohms/km) (Amp) 75 ◦C Mole 4.5 2.78 70 Squirrel 6.33 1.394 107 Weasel 7.77 0.9291 138 Rabbit 10.05 0.5524 190 Racoon 12.27 0.3712 244 Dog 14.15 0.2792 291 Wolf 18.13 0.1871 405 Lynx 19.53 0.161 445 Panther 21 0.139 487 Zebra 28.62 0.06869 737 Dear 29.89 0.06854 756 Moose 31.77 0.05596 836 Bersimis 35.04 0.04242 998 18 School of Engineering
TL Parameters - Resistance
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TL conductor resistance depends on factors such as:
Conductor geometry
The frequency of the AC current
Conductor proximity to other current-carrying conductors
Temperature
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Resistance
풍 The DC resistance of a conductor at a temperature T is given by: 푹(푻) = 흆 푻 푨 where
T = conductor resistivity at temperature T l = the length of the conductor A = the current-carry cross-sectional area of the conductor
푷 The AC resistance of a conductor is given by: 푹 = 풍풐풔풔 풂풄 푰ퟐ where
Ploss – real power dissipated in the conductor in watts I – rms conductor current 21 School of Engineering
TL Conductor Resistance depends on:
Spiraling:
The purpose of introducing a steel core inside the stranded aluminum conductors is to obtain a high strength-to-weight ratio. A stranded conductor offers more flexibility and easier to manufacture than a solid large conductor. However, the total resistance is increased because the outside strands are larger than the inside strands on account of the spiraling.
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The layer resistance-per-length of each spirally wound conductor depends on its total length as follows:
흆 ퟏ ퟐ 푹풄풐풏풅 = ퟏ + (흅 ) Ω/m 푨 풑풄풐풏풅 where
푅푐표푛푑 - resistance of the wound conductor (Ω)
1 1 + (휋 )2 - Length of the wound conductor (m) 푃푐표푛푑
푙 푡푢푟푛 p푐표푛푑 = - relative pitch of the wound conductor (m) 2푟푙푎푦푒푟
푙푡푢푟푛 - length of one turn of the spiral (m)
2푟푙푎푦푒푟 - Diameter of the layer (m) 23 School of Engineering
Frequency:
When voltages and currents change in time, current flow (i.e. current density) is not uniform across the diameter of a conductor.
Thus the ‘effective’ current-carrying cross-section of a conductor with AC is less than that for DC.
This phenomenon is known as skin effect.
As frequency increases, the current density decreases from that at the surface of the conductor to that at the center of the conductor. 24 School of Engineering
Frequency cont.
The spatial distribution of the current density in a particular conductor is additionally altered (similar to the skin effect) due to currents in adjacent current-carrying conductors.
This is called the ‘Proximity Effect’ and is smaller than the skin-effect.
Using the DC resistance as a starting point, the effect of AC on the overall cable resistance may be accounted for by a correction factor k.
k is determined by E&M analysis. For 60 Hz, k is estimated around 1.02
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Skin Effect
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Temperature:
The resistivity of conductors is a function of temperature.
For common conductors (Al & Cu), the conductor resistance increases ~ linearly with temperature
휌 푇 = 휌0 1 + 훼 푇 − 푇표
the temperature coefficient of resistivity
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TL Parameters - Conductance
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Conductance
Conductance is associated with power losses between the conductors or between the conductors and ground.
Such power losses occur through leakage currents on insulators and via a corona
Leakage currents are affected by: Contaminants such dirt and accumulated salt accumulated on insulators. Meteorological factors such as moisture
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Conductance
Corona loss occurs when the electric field at the surface of a conductor causes the air to ionize and thereby conduct.
Corona loss depends on: Conductor surface irregularities Meteorological conditions such as humidity, fog, and rain
Losses due to leakage currents and corona loss are often small compared to direct I2R losses on TLs and are typically neglected in power flow studies.
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Corona loss
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Corona loss
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TL Parameters - Inductance
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Inductance is defined by the ratio of the total magnetic flux flowing through (passing through) an area divided by the current producing that flux
Φ = B ∙ d푠 ∝ 퐼 , 푂푏푠푒푟푣푎푡푖표푛 S Φ L = , 퐷푒푓푖푛푖푡푖표푛 퐼
Inductance is solely dependent on the geometry of the arrangement and the magnetic properties (permeability ) of the medium. Also, B = H where H is the mag field due to current flow only. 34 School of Engineering
Magnetic Flux
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Inductance
When a current flows through a conductor, a magnetic flux is set up which links the conductor.
The current also establishes a magnetic field proportional to the current in the wire.
Due to the distributed nature of a TL we are interested in the inductance per unit length (H/m). 36 School of Engineering
Inductance - Examples
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Inductance of a Single Wire
The inductance of a magnetic circuit that has a constant permeability (휇) can be obtained as follows:
The total magnetic field Bx at x via Amperes law:
휇 푥 퐵 = 표 퐼 (푊푏/푚) 푥 2휋푟2
The flux d through slice dx is given by:
푑Φ = 퐵푥 푑푎 = 퐵푥 푙 푑푥 (푊푏) 38 School of Engineering
The fraction of the current I in the wire that links the area da = dxl is:
x 2 N = f r
The total flux through the wire is consequently given by:
푟 푟 2 푥 휇표 푥 퐼 Φ = 푑Φ = 2 푙푑푥 0 0 푟 2휋푟
Integrating with subsequent algebra to obtain inductance-per-m yields:
−7 퐿푤푖푟푒 −푠푒푙푓 = 0.5 ∙ 10 (퐻/푚) 39 School of Engineering
Inductance per unit length (H/m) for two Wires (Cables)
To calculate the inductance per unit length (H/m) for two-parallel wires the flux through area S must be calculated. 40 School of Engineering
The inductance of a single phase two-wire line is given by: D L = 4 ∙ 10−7 ln (H/m) r′ 1 ′ ′ ′ − where 푟 = 푟푥 = 푟푦 = 푒 4 푟푥
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The inductance per phase of a 3-phase 3 wire line with equal spacing is given by:
D L = 2 ∙ 10−7 ln (H/m) per phase a r′
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L of a Single Phase 2-conductor Line Comprised of Composite Conductors
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−7 퐷푥푦 퐿푥 = 2 ∙ 10 ln 퐷푥푥
where
푁 푀 푁 푀 푀푁 푁2 퐷푥푦 = 퐷푘푚 푎푛푑 퐷푥푥 = 퐷푘푚 푘=1 푚=1′ 푘=1 푚=1
Dxy is referred to as GMD = geometrical mean distance between conductors x & y
Dxx is referred to as GMR = geometrical mean radius of conductor x
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A similar Ly express exists for conductor y.
LTotal = Lx + Ly (H/m)
Great NEWs!!!!!
Manufacturers often calculate these inductances for us for various cable arrangements
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Long 3-phase TLs are sometimes transposed for positive sequence balancing.
This is cleverly called ‘transposition’
The inductance (H/m) of a completely transposed three phase line may also be calculated
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−7 퐷푒푞 퐿푎 = 2 ∙ 10 ln 퐷푠
3 퐷푒푞 = 퐷12퐷13퐷23
La has units of (H/m)
Ds is the conductor GMR for stranded conductors or r’ for solid conductors
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In HV TLs conductors are often arranged in one of the following 3 standard configurations
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−7 퐷푒푞 퐿푎 = 2 ∙ 10 ln 퐷푆퐿
La again has units of (H/m)
If bundle separation is large compared to the bundle size, then Deq ~ the center-to- center distance of the bundles
4 2 2-conductor bundle: 퐷푆퐿 = 퐷푆 푑 = 퐷푆푑
9 3 3 2 3-conductor bundle: 퐷푆퐿 = 퐷푆 푑푑 = 퐷푆푑
16 4 4 3 4-conductor bundle: 퐷푆퐿 = 퐷푆 푑푑푑 2 = 1.091 퐷푆푑 49 School of Engineering
TL Parameters - Capacitance
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휀 퐸 ∙ 푑푠 휀 Q = C V C = 푆 RC = 휎 − 푙 퐸 ∙ 푑푙 School of Engineering
Capacitance
A capacitor results when any two conductors are separated by an insulating medium.
Conductors of an overhead transmission line are separated by air – which acts as an insulating medium – therefore they have capacitance.
Due to the distributed nature of a TL we are interested in the capacitance per unit length (F/m).
Capacitance is solely dependent on the geometry of the arrangement and the electric properties (permittivity ) of the medium
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Calculating Capacitance
To calculate the capacitance between conductors we must calculate two things:
The flux of the electric field E between the two conductors
The voltage V (electric potential) between the conductors
Then we take the ratio of these two quantities 53 School of Engineering
Since conductors in a power system are typically cylindrical, we start with this geometry for determining the flux of the electric field and potential (reference to infinity)
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Gauss’s Law may be used to calculate the electric field E and the potential V for this geometry:
푞 퐸 = (푉/푚) 푥 2휋휀푥
The potential (voltage) of the wire is found using this electric field:
푞 퐷2 푑푥 푞 퐷 푉 = = 푙푛 2 푉표푙푡푠 12 2휋휀 푥 2휋휀 퐷 퐷1 1
The capacitance of various line configurations may be found using these two equations 55 School of Engineering
Capacitance - Examples
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Capacitance of a single-phase 2-Wire Line
The line-to-neutral capacitance is found to be:
2휋휀 퐶푛 = 퐷 (퐹/푚) 푙푛 푟
The same Cn is also found for a 3-phase line-to-neutral arrangement School of Engineering
The Capacitance of Stranded Conductors
e.g. a 3-phase line with 2 conductors per phase
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Direct analysis yields
2휋휀 퐶 = (퐹/푚) 푎푛 퐷 푙푛 푒푞 푟 where
3 퐷푒푞 = 퐷푎푏퐷푎푐퐷푏푐
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The Capacitance of a Transposed Line
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The Capacitance of a Transposed Line
2휋휀 퐶 = (퐹/푚) 푎푛 퐷 푙푛 푒푞 퐷푆퐶 where
3 퐷푒푞 = 퐷푎푏퐷푎푐퐷푏푐
2-conductor bundle: 퐷푆퐶 = 푟 푑
3 2 3-conductor bundle: 퐷푆퐶 = 푟 푑
4 3 4-conductor bundle: 퐷푆퐶 = 1.091 푟 푑 61 School of Engineering
TLs Continued
Dr. Greg Mowry Annie Sebastian Marian Mohamed
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Maxwell’s Eqns. & the Telegraph Eqns.
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Maxwell’s Equations & Telegraph Equations
The ‘Telegrapher's Equations’ (or) ‘Telegraph Equations’ are a pair of coupled, linear differential equations that describe the voltage and current on a TL as functions of distance and time.
The original Telegraph Equations and modern TL model was developed by Oliver Heaviside in the 1880’s.
The telegraph equations lead to the conclusion that voltages and currents propagate on TLs as waves. 4 School of Engineering
Maxwell’s Equations & Telegraph Equations
The Transmission Line equation can be derived using Maxwell’s Equation.
Maxwell's Equations (MEs) are a set of 4 equations that describe all we know about electromagnetics.
MEs describe how electric and magnetic fields propagate, interact, and how they are influenced by other objects.
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Time for History
James Clerk Maxwell [1831-1879] was an Einstein/Newton- level genius who took the set of known experimental laws (Faraday's Law, Ampere's Law) and unified them into a symmetric coherent set of Equations known as Maxwell's Equations.
Maxwell was one of the first to determine the speed of propagation of electromagnetic (EM) waves was the same as the speed of light - and hence concluded that EM waves and visible light were really descriptions of the same phenomena.
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Maxwell’s Equations – 4 Laws
Gauss’ Law: The total of the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity. The electric flux through an area is defined as the electric field multiplied by the area of the surface projected in a plane perpendicular to the field. 흆 휵 ∙ 푫 = 흐
The ‘no name’ ME: The net magnetic flux through any closed surface is zero.
훁 ∙ 푩 = ퟎ 7 School of Engineering
Faraday’s law: Faraday's ‘law of induction’ is a basic law of electromagnetism describing how a magnetic field interacts over a closed path (often a circuit loop) to produce an electromotive force (EMF; i.e. a voltage) via electromagnetic induction.
흏푩 휵 푿 푬 = − 흏풕
Ampere’s law: describes how a magnetic field is related to two sources; (1) the current density J, and (2) the time-rate-of-change of the displacement vector D.
흏푫 휵 푿 푯 = + 푱 흏풕 8 School of Engineering
The Constitutive Equations
Maxwell’s 4 equations are further augmented by the ‘constitutive equations’ that describe how materials interact with fields to relate (B & H) and (D & E).
For simple linear, isotropic, and homogeneous materials the constitutive equations are represented by:
푩 = 흁 푯 흁 = 흁풓 흁풐
푫 = 휺 푬 흐 = 휺풓 휺풐 9 School of Engineering
The Telegraph Eqns.
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TL Equations
TL equations can be derived by two methods: Maxwell’s equations applied to waveguides Infinitesimal analysis of a section of a TL operating in ACSS with LCRG parameters (i.e. use phasors)
The TL circuit model is an infinite sequence of 2-port components; each being an infinitesimally short segment of the TL. 11 School of Engineering
TL Equations
The LCRG parameters of a TL are expressed as per- unit-length quantities to account for the experimentally observed distributed characteristics of the TL
R’ = Conductor resistance per unit length, Ω/m
L’ = Conductor inductance per unit length, H/m
C’ = Capacitance per unit length between TL conductors, F/m
G’ = Conductance per unit length through the insulating medium between TL conductors, S/m 12 School of Engineering
The Infinitesimal TL Circuit
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The Infinitesimal TL Circuit Model
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Transmission Lines Equations
The following circuit techniques and laws are used to derive the lossless TL equations
KVL around the ‘abcd’ loop
KCL at node b
휕i V = L OLL for an inductor L 휕t
휕v I = C OLC for a capacitor C 휕t
Set 푅′= 퐺′= 0 for the Lossless TL case 15 School of Engineering
Apply KVL around the “abcd” loop: 휕퐼(푧) 푉(z) = V(z+∆z) +푅′∆z I(z) + 퐿′∆z 휕푡
Divide all the terms by ∆z : 푉 푧 푉 푧+∆푧 휕퐼(푧) = + 푅′퐼 푧 + 퐿′ ∆푧 ∆푧 휕푡
As ∆z 0 in the limit and R’ = 0 (lossless case): 휕푉(푧) 휕퐼(푧) = – 푅′퐼 푧 + 퐿′ 휕푧 휕푡
흏푽(풛) 흏푰(풛) = –푳′ (Eq.1) 흏풛 흏풕 School of Engineering
Apply KCL at node b: 휕푉(푧+∆푧) 퐼(z) = I(z+∆z) +퐺′∆z V(z+∆z) + 퐶′∆z 휕푡
Divide all the terms by ∆z: 퐼 푧 퐼 푧+∆푧 휕푉(푧+∆푧) = + 퐺′푉 푧 + ∆푧 + 퐶′ ∆푧 ∆푧 휕푡
As ∆z 0 in the limit and G’ = 0 (lossless case): 휕퐼(푧) 휕푉(푧) = – 퐺′푉 푧 + 퐶′ 휕푧 휕푡
흏푰(풛) 흏푽(풛) = – 푪′ (Eq.2) 흏풛 흏풕 School of Engineering
Summarizing
휕 푉 푧 ,푡 휕 퐼 푧 ,푡 (Eqn. 1) = − 퐿′ 휕푧 휕푡
휕 퐼 푧 ,푡 휕 푉 푧 ,푡 (Eqn. 2) = − 퐶′ 휕푧 휕푡
Equations 1 & 2 are called the “Coupled First Order Telegraph Equations” 18 School of Engineering
Differentiating (Eq.1) wrt ‘z’ yields: 휕2푉(푧) 휕2퐼(푧) = – 퐿′ (Eq.3) 휕푧2 휕푧휕푡
Differentiating (2) wrt ‘t’ yields: 휕2퐼(푧) 휕2푉(푧) = – 퐶′ (Eq.4) 휕푧휕푡 휕푡2
Plugging equation 4 into 3:
ퟐ ퟐ 흏 푽(풛) ′ ′ 흏 푽(풛) = 푳 푪 (Eq.5) 19 흏풛ퟐ 흏풕ퟐ School of Engineering
Reversing the differentiation sequence Differentiating (Eq.1) wrt ‘t’ yields: 휕2푉(푧) 휕2퐼(푧) = – 퐿′ (Eq.6) 휕푧휕푡 휕푡2
Differentiating (2) wrt ‘z’ yields: 휕2퐼(푧) 휕2푉(푧) = – 퐶′ (Eq.7) 휕푧2 휕푧휕푡
Plugging equation 6 into 7:
흏ퟐ푰(풛) 흏ퟐ푰(풛) = 푳′푪′ (Eq.8) 흏풛ퟐ 흏풕ퟐ 20 School of Engineering
Summarizing
휕2V(z) 휕2V(z) (Eqn. 5) = L′C′ 휕z2 휕t2
휕2I(z) 휕2I(z) (Eqn. 8) = L′C′ 휕z2 휕t2
Equations 5 & 8 are the v(z,t) and i(z,t) TL wave equations 21 School of Engineering
What is really cool about Equations 5 & 8 is that the solution to a wave equation was already known; hence solving these equations was straight forward.
The general solution of the wave equation has the following form:
f z, t = f(z ± ut)
where the ‘wave’ propagates with Phase Velocity 푢
ω ω 1 u = = = β ω L′C′ L′C′
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The solution of the general wave equation, f(z,t), describes the shape of the wave and has the following form:
f z, t = f(z − ut) or f z, t = f(z + ut)
The sign of 푢푡 determines the direction of wave propagation:
푓 푧 − 푢푡 Wave traveling in the +z direction
푓 푧 + 푢푡 Wave traveling in the –z direction 23 School of Engineering
The ‘+z’ Traveling WE Solution 푓 푧, 푡 = 푓(푧 − 푢푡)
푓 푧 − 푢푡) = 푓(0
푧 − 푢푡 = 0
푧 = 푢푡 24 School of Engineering
The argument of the wave equation solution is called the phase.
Hence the culmination of all of this is that, “Waves propagate by phase change”
One other key point, the form of the voltage WE and current WE are identical. Hence solutions to each must be of the same form and in phase.
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Wave Equation z increases as t increase for a constant phase
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TL Solutions
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TL Wave Equations - Recap
휕2V(z) 휕2V(z) (Eqn. 5) = L′C′ 휕z2 휕t2
휕2I(z) 휕2I(z) (Eqn. 8) = L′C′ 휕z2 휕t2
Equations 5 & 8 are the v(z,t) and i(z,t) TL wave equations 28 School of Engineering
ACSS TL Analysis Recipe
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Initial Steps
Typically we will know or are given the source voltage. For example, it could be:
7 ° 풗풔풔 풕 = ퟏퟎ cos 10 푡 + 30 풗풐풍풕풔
From the time function we form the source phasor & know the frequency that the TL operates at (in ACSS operation):
훚 = ퟏퟎퟕ 풔−ퟏ
풋ퟑퟎ° 푽풔풔 = ퟏퟎ 풆 풗풐풍풕풔 30 School of Engineering
ACSS TL Analysis
Under ACSS conditions the voltage and current solutions to the WE may be expressed as:
+ 푗(휔푡−γ푧) − 푗(휔푡+γ푧) 푉 푧, 푡 = 푅푒 푉0 푒 + 푉0 푒
+ −푗γ푧 − 푗γ푧 푉 푧 = 푉0 푒 + 푉0 푒 Time Harmonic Phasor Form
+ 푗(휔푡−γ푧) − 푗(휔푡+γ푧) I 푧, 푡 = 푅푒 퐼0 푒 + 퐼0 푒
+ −푗γ푧 − 푗γ푧 푉+ −푗γ푧 푉− 푗γ푧 I 푧 = 퐼0 푒 + 퐼0 푒 = 푒 + 푒 Time Harmonic Phasor Form 푍0 푍0 School of Engineering
ACSS TL Analysis Recipe
Step 1: Assemble 푍0 , 푍퐿, 푙, 푎푛푑 γ
Step 2: Determine Г퐿
Step 3: Determine 푍푖푛
Step 4: Determine 푉푖푛
+ Step 5: Determine 푉0
Step 6: Determine 푉퐿
Step 7: Calculate 퐼퐿 and 푃퐿 32 School of Engineering
Step 1: Assemble 풁ퟎ , 풁푳, 풍, , and 휸
Characteristic Impedance: ′ + − 푳 푽풐 푽풐 풁풐 = ′ = + = − − 푪 푰풐 푰풐 Load Impedance: + − 푽푳 푽ퟎ + 푽ퟎ 풁푳 = = + − 풁ퟎ 푰푳 푽ퟎ − 푽ퟎ
Complex Propagation Constant: ( = 0 for lossless TL) 휸 = 휶 + 풋휷 = 풋흎 푳′푪′
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Additional Initial Results
Phase velocity uP: 훚 ퟏ 퐮퐏 = = 훃 퐋′ 퐂′
Identity: 퐋′ 퐂′ = 훍 훆
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Step 2: Determine Г푳
Reflection Coefficient: − − 푉0 퐼0 Г퐿 = + = − + 푉0 퐼0 푍퐿−푍0 Г퐿 = 푍퐿+푍0 푍퐿 푧 퐿 = 푍0 풛푳 − ퟏ Г푳 = 풛푳 + ퟏ
Reflection Coefficient at distance z (from load to source = -l):
− 휸풛 푽풐 풆 풋ퟐ휷풛 Г 풛 = + −휸풛 = Г푳풆 푽풐 풆 35 School of Engineering
Additional Step-2 Results
VSWR:
ퟏ + 횪 퐕퐒퐖퐑 = 푳 ퟏ − 횪푳
Range of the VSWR:
ퟎ < 푽푺푾푹 < ∞
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Step 3: Determine 풁풊풏
Input Impedance: 푽풊풏 풛 = −풍 풁푳 + 풁ퟎ퐭퐚퐧퐡(휸풍) 풁풊풏 = = 풁ퟎ 푰풊풏 풛 = −풍 풁ퟎ + 풁푳풕풂풏풉(휸풍)
For a lossless line tanh γl = j tan(βl):
풁푳 + 풋풁ퟎ풕풂풏(휷풍) 풁풊풏 = 풁ퟎ 풁ퟎ + 풋풁푳풕풂풏(휷풍)
At any distance 푧1 from the load:
풁푳 + 풋풁ퟎ퐭퐚퐧(휷풛ퟏ) 풁(풛ퟏ) = 풁ퟎ 풁ퟎ + 풋풁푳풕풂풏(휷풛ퟏ) 37 School of Engineering
Step 4: Determine 푽풊풏
풁풊풏 푽풊풏 = 푽푺푺 = 푽풊풏 풛 = −풍 풁풊풏 + 풁푺
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+ Step 5: Determine 푽ퟎ
+ −푗훽푧 − 푗훽푧 + −푗훽푧 푗훽푧 푉 푧 = 푉0 푒 + 푉0 푒 = 푉0 (푒 +Г퐿 푒 )
+ 푗훽푙 −푗훽푙 푉푖푛 = 푉푇퐿 푧 = −푙 = 푉0 (푒 +Г퐿 푒 )
+ 푽풊풏 푽ퟎ = 풋휷풍 −풋휷풍 (풆 +Г푳 풆 )
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Step 6: Determine 푽푳
Voltage at the load, VL:
+ 푽푳 = 푽푻푳 풛 = ퟎ = 푽풐 ퟏ + 횪푳
where:
+ 푽풊풏 푽ퟎ = 풋휷풍 −풋휷풍 (풆 +Г푳 풆 )
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Step 7: Calculate 푰푳 퐚퐧퐝 푷푳
푽푳 푰푳 = 풁푳
ퟏ 푷 = 푷 = 푰ퟐ푹풆(풁 ) 풂풗품 푳 ퟐ 푳 푳
∗ 푰풏 품풆풏풆풓풂풍, 푺푳 = 푽푳 푰푳 = 푷푳 + 풋 푸푳
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Final Steps
After we have determined VL in phasor form, we might be interested in the actual time function associated with VL; i.e. vL(t). It is straight forward to convert the VL phasor back to the vL(t) time-domain form; e.g. suppose:
−풋ퟔퟎ° 푽푳 = ퟓ 풆 풗풐풍풕풔
Then with 훚 = ퟏퟎퟕ 풔−ퟏ
7 ° 풗푳 풕 = ퟓ cos 10 푡 − 60 풗풐풍풕풔 42 School of Engineering
Surge Impedance Loading (SIL)
43 Lossless TL with VS fixed for line-lengths < /4 School of Engineering
2-Port Analysis
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Two Port Network
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A two-port network is an electrical ‘black box’ with two sets of terminals.
The 2-port is sometimes referred to as a four terminal network or quadrupole network.
In a 2-port network, port 1 is often considered as an input port while port 2 is considered to be an
output port. 46 School of Engineering
A 2-port network model is used to mathematically analyze electrical circuits by isolating the larger circuits into smaller portions.
The 2-port works as “Black Box” with its properties specified by a input/output matrix.
Examples: Filters, Transmission Lines, Transformers, Matching Networks, Small Signal Models 47 School of Engineering
Characterization of 2-Port Network
2-port networks are considered to be linear circuits hence the principle of superposition applies.
The internal circuits connecting the 2-ports are assumed to be in their zero state and free of independent sources.
2-Port Network consist of two sets of input & output
variables selected from the overall V1, V2, I1, I2 set: 2 Independent (or) Excitation Variables 48 2 Dependent (or) Response Variables School of Engineering
Characterization of 2-Port Network
Uses of 2-Port Network:
Analysis & Synthesis of Circuits and Networks.
Used in the field of communications, control system, power systems, and electronics for the analysis of cascaded networks.
2-ports also show up in geometrical optics, wave-guides, and lasers
By knowing the matric parameters describing the 2-port network, it can be considered as black-box when embedded within a larger network. 49 School of Engineering
2-Port Networks & Linearity
Any linear system with 2 inputs and 2 outputs can always be expressed as:
푦1 푎11 푎12 푥1 = 푎 푎 푦2 21 22 푥2
50 2-Port Networks http://www.globalspec.com/reference/9518/348308/chapter-10-two-port-networks
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Review
A careful review of the preceding TL analysis reveals that all we
really determined in ACSS operation was Vin & Iin and VL & IL.
This reminds us of the exact input and output form of any linear system with 2 inputs and 2 outputs which can always be expressed as:
푦1 푎11 푎12 푥1 = 푎 푎 ∴ 푦2 21 22 푥2 52 School of Engineering
Transmission Line as Two Port Network
I1 = Iin I2 = IL
ZS + + A B + V V = V V = V Z S - 1 in 2 L L C D - -
Source 2-Port Network Load School of Engineering
Representation of a TL by a 2-port network
Nomenclature Used:
VS – Sending end voltage
IS – Sending end current
VR – Receiving end Voltage
IR – Receiving end current
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Representation of a TL by a 2-port network
Relation between sending end and receiving end is given as:
푉푆 = 퐴 푉푅 + 퐵 퐼푅 퐼푆 = 퐶 푉푅 + 퐷 퐼푅
In Matrix Form
푉푆 퐴 퐵 푉푅 = 퐼푆 퐶 퐷 퐼푅
퐴퐷 − 퐵퐶 = 1
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4 Cases of Interest
Case 1: Short TL
Case 2: Medium distance TL
Case 3: Long TL
Case 4: Lossless TL 56 School of Engineering
ABCD Matrix- Short Lines
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Short TL Less than ~80 Km shunt Admittance is neglected
Apply KVL & KCL on abcd loop:
VS = VR + 푍IR
IS = IR
ABCD Matrix for short lines: 1 푍 푉푆 = 푉푅 퐼푆 0 1 퐼푅
ABCD Parameters: A = D = 1 per unit B = 푍 Ω C = 0 58 School of Engineering
ABCD Matrix- Medium Lines
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Medium TL
Ranges from ~ 80 to 250 Km Nominal symmetrical-π circuit model used
Apply KVL & KCL on abcd loop: 푍 푌 V = 1 + V + ZI S 2 R R 푌푍 푌푍 I = Y 1 + V + 1 + I s 4 R 2 R
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Medium TL
ABCD Matrix for medium lines:
푌 푍 1 + 푍 푉 2 푉 푆 = 푅 퐼 푌 푍 푌 푍 퐼 푆 푌 1 + 1 + 푅 4 2
ABCD Parameters: 푌푍 A = D = 1 + per unit 2 B = Z Ω 푌푍 61 C = Y 1 + Ω-1 4 School of Engineering
ABCD Matrix- Long Lines
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Long TL
More than ~ 250 Km and beyond Full TL analysis
ZC = Z0
ABCD Matrix for long lines:
cosh(훾푙) 푍 sinh(훾푙) 푉 퐶 푉 푆 = 1 푅 퐼푆 sinh(훾푙) cosh(훾푙) 퐼푅 푍퐶
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ABCD Matrix- Lossless Lines R = G = 0
퐿′ 푍 = 0 퐶′ γ = 푗ω 퐿′퐶′ = 푗β
ABCD Matrix for lossless lines: cos(훽푙) 푗 푍 sin(훽푙) 푉 퐶 푉 푆 = 푗 푅 퐼푆 sin(훽푙) cos(훽푙) 퐼푅 푍퐶
ABCD Parameters: 퐴 = 퐷 = cosh 훾푙 = cosh(푗β푙) = cos(β푙) per unit
퐵 = 푍0 sinh γ푙 = 푗푍0sin(β푙) Ω sinh(훾푙) sin(훽푙) 퐶 = = 푗 S 64 푍0 푍0 School of Engineering
ABCD Matrix-Summary
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A Brief Recipe on ‘How to do’ Power Flow Analysis (PFA)
Cooking with Dr. Greg Mowry School of Engineering
Steps
1. Represent the power system by its one-line diagram
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Steps Cont.
2. Convert all quantities to per unit (pu)
2-Base pu analysis:
SBase kVA, MVA, …
VBase V, kV, …
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Steps Cont.
3. Draw the impedance diagram
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Steps Cont.
4. Determine the YBUS
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Steps Cont.
5. Classify the Buses; aka ‘bus types’
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Steps Cont.
6. Guess starting values for the unknown bus parameters; e.g.
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Steps Cont.
7. Using the power flow equations find approximations for the real & reactive power using the initially guessed values and the known values for voltage/angles/admittance at the various buses. Find the difference in these calculated values with the values that were actually given; i.e. form P & Q.
(Δ value ) = (Given Value) – (Approximated Value)
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Steps Cont.
8. Write the Jacobian Matrix for the first iteration of the Newton Raphson Method. This will have the form:
[Δvalues] = [Jacobian Matrix] * [ for Unknown Parameters]
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Steps Cont.
9. Solve for the unknown differences by inverting the Jacobian and multiplying & form the next guess:
(Unknown Value)new = (Unknown Value)old + Solved (Δ Value) 11 School of Engineering
Steps Cont.
10. Repeat steps 7 – 9 iteratively until an accurate value for the unknown differences as they 0. Then solve for all the other unknown parameters.
Once these iterations converge (e.g. via NR), we now have
a fully solved power system with respect to Sj & Vj at all of the system buses!!
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Power Flow Analysis (PFA)
Dr. Greg Mowry Annie Sebastian Marian Mohamed
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Introduction
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Power Flow Analysis
In modern power systems, bus voltages and power flow are controlled.
Power flow is proportional to v2, therefore controlling both bus voltage & power flow is a nonlinear problem.
In power engineering, the power-flow study, or load-flow study, is a numerical analysis of the flow of electric power in an interconnected system.
A power-flow study usually a steady-state analysis that uses simplified notation such as a one-line diagram and per-unit system, and focuses on various aspects of AC power parameters, such as voltages, voltage angles, real power and reactive power. 3 School of Engineering
Power Flow Analysis - Intro
PFA is very crucial during the power-system planning phases as well as during periods of expansion and change for meeting present and future load demands.
PFA is very useful (and efficient) at determining power flows and voltage levels under normal operating conditions.
PFA provides insight into system operation and optimization of control settings, which leads to maximum capacity at lower the operating costs.
PFA is typically performed under ACSS conditions for 3-phase power systems 4 School of Engineering
PFA Considerations
1. Generation Supplies the demand (load) plus losses.
2. Bus Voltage magnitudes remain close to rated values.
3. Gens Operate within specified real & reactive power limits
4. Transmission lines and Transformers are not overloaded. 5 School of Engineering Load Flow Studies (LFS)
Power-Flow or Load-Flow software (e.g. CYME, Power World, CAPE) is used for investigating power system operations.
The SW determines the voltage magnitude and phase at each bus in a power system under balanced three-phase steady state conditions.
The SW also computes the real and reactive power flows (P and Q respectively) for all loads, buses, as well as the equipment losses.
The load flow is essential to decide the best operation of existing system, for planning the future expansion of the system, and designing new power system.
6 School of Engineering LFS Requirements
Representation of the system by a one-line diagram.
Determining the impedance diagram using information in the one- line diagram.
Formulation of network equations & PF equations.
Solution of these equations.
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Conventional Circuit Analysis vs. PFA
Conventional nodal analysis is not suitable for power flow studies as the input data for loads are normally given in terms of power and not impedance.
Generators are considered power sources and not voltage or current sources
The power flow problem is therefore formulated as a set of non- linear algebraic equations that require numerical analysis for the solution. 8 School of Engineering Bus
A ‘Bus’ is defined as the meeting point (or connecting point) of various components.
Generators supply power to the buses and the loads draw power from buses.
In a power system network, buses are considered as nodes and hence the voltage is specified at each bus
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The Basic 2-Bus Analysis The Foundational Heart of PFA
Quick Review
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Motivation
The basic 2-Bus analysis must be understood by all power engineers.
The 2-Bus analysis is effectively the ‘Thevenin Equivalent” of all power systems and power flow analysis.
Many variant exist, by my count, at least a dozen. We will explore the basic 2-Bus PFA that models transmission systems. 11 School of Engineering
The Basic 2-Bus Power System
1 2 I j X + -
푗훿 ° 푉 푆 = 푉푆 ∠훿 = 푉푅푒 푉푅 = 푉푅 ∠0 = 푉푅
S = ‘Send’ R = ‘Receive’ 12 School of Engineering
The Basic 2-Bus Power System
1 2 I ~ j X ~ + -
푗훿 ° 푉 푆 = 푉푆 ∠훿 = 푉푅푒 푉푅 = 푉푅 ∠0 = 푉푅
S = ‘Send’ R = ‘Receive’ 13 School of Engineering
2-Bus Phasor Diagram
VR the Reference
VS j X I
O V R I 90
j X I I due to j 14 School of Engineering
Discussion
1) VS leads VR by
2) KVL VS – j X I = VR
푉 − 푉 ∴ 퐼 = 푆 푅 푗푋 15 School of Engineering
Discussion
3) At the receive end,
∗ 푆푅 = 푃푅 + 푗푄푅 = 푉푅 퐼
푉 ∠훿 − 푉 ∗ 푆 = 푉 푆 푅 푅 푅 푗푋
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Discussion
4) Therefore the complex power received is (after algebra):
푉 푉 푉 푉 푉2 푆 = 푆 푅 sin 훿 + 푗 푆 푅 cos 훿 − 푅 푅 푋 푋 푋
푆푅 = 푃푅 + 푗푄푅
푗훿 Recall 푉 푆 = 푉푆∠훿 = 푉푆푒 = 푉푆 cos 훿 + 푗 sin 훿 17 School of Engineering
Discussion
5) Or:
푉 푉 푃 = 푆 푅 sin 훿 푅 푋
푉 푉 푉2 푄 = 푆 푅 cos 훿 − 푅 푅 푋 푋
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Discussion
6) Note that if VS ~ VR ~ V then:
푉2 푃 ≈ 훿 푅 푋
푉 푄 ≈ ∆푉 푅 푋
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2-Bus, Y, and PFA Lead-in
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Two Bus System – arbitrary phase angles
21 School of Engineering Two Bus System
The complex AC power flow at the receiving end bus can be calculated as follows:
∗ 푺풓 = 푽풓 푰
Similarly, the power flow at the sending end bus is:
∗ 푺풔 = 푽풔 푰 where
푗휑1 푉 푠 = 푉푠 푒 is the voltage and phase angle at the sending bus. 푗휑2 푉 푟 = 푉푟 푒 is the voltage and phase angle at the receiving bus. Z is the complex impedance of the TL. 22 School of Engineering Two Bus System
푽 − 푽 퐈 = 풔 풓 풁
In power flow calculations, it is more convenient to use admittances rather than impedances
풀 = 풁−ퟏ
푰 = 풀 푽풔 − 푽풓
This 2-bus equation forms the basis of PF calculations. 23 School of Engineering The Admittance Matrix
The use of admittances greatly simplifies PFA
e.g. an open circuit between two buses (i.e. no connection) results in Y = 0 (which is computationally easy to deal with) rather than Z = ∞ (which is computationally hard to deal with).
Vij is the voltage at bus i wrt bus j; bus i positive, bus j negative
Iab represents the current flow from bus a to bus b
Recall that a ‘node voltage’ (bus voltage) is the voltage at that node (Bus) with respect to the reference. 24 School of Engineering Admittance Matrix Method (1)
NV analysis results in n-1 node equations (bus equations) which is analytically very efficient for systems with large n.
Given systems with n large, an efficient and automatable method is needed/required for PFA. NVM and Y-matrix methods are so.
In circuit analysis, series circuit elements are typically combined to form a single impedance to simplify the analysis. This is generally not done in PFA. 25 School of Engineering Admittance Matrix Method (2)
In PFA, combining series elements is not typically done as we seek to keep track of the impedances of various assets and their power flows; e.g. in: i. Generators ii. TLs iii. XFs iv. FACTs
Voltages sources (e.g. generators) have an associated impedance (the Thevenin impedance) that is typically inductive in ACSS
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Y-Matrix Example
27 School of Engineering Example Circuit
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Step 1: Identify nodes (buses), label them, and select one of them as the reference or ground.
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Step 2 & 3: (2) Replace voltage sources and their Thevenin impedance with their Norton equivalent. (3) Replace impedance with their associated admittances.
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Step 4: Form the system matrix
풀푩풖풔 ∗ 푽 = 푰 (Eqn-1) where,
Ybus = Bus Admittance matrix of order (n x n); known V = Bus (node) voltage matrix of order (n x 1); unknown
I = Source current matrix of order (n x 1); known 31 School of Engineering
Step 5: Solve the system matrix for bus voltages
−ퟏ 푽 = 풀푩풖풔 ∗ 푰 (Eqn-2)
Note: at this point the various known bus voltages (amplitude and phase) may be used to calculate currents and power flow.
32 School of Engineering Stuffing the Y-Matrix
th Y-matrix elements are designated as 푌푘푛 where k refers to the k bus and n refers to the nth bus.
Diagonal elements 푌푘푘:
th 푌푘푘 = sum of all of the admittances connected to the k bus
Off-diagonal elements 푌푘푛:
푌푘푛 = 푌푛푘 = -(sum of the admittances in the branch connecting bus k to bus n) 33 School of Engineering Y-Matrix Terminology
푌푘푘 , the main diagonal elements of Y, are referred to as the:
“Self-admittance”; or “Driving point admittance” of bus k
푌푘푛 , the off diagonal elements of Y, are referred to as the:
“Mutual-admittance”; or “Transfer admittance” between bus k and bus n 34 School of Engineering Y-Matrix Notes
If there is no connection between bus k and bus n, then 푌푘푛 = 0 in the Y-matrix; hence there are 2 zeros in the Y-matrix for this situation since 푌푘푛 = 푌푛푘.
If a new connection is made between 2 buses k & n in a network, then only 4 elements of the Y matrix change, all of the others remain the same. The elements that change are:
푌푘푘 푌푛푛 푌푘푛 = 푌푛푘 35 School of Engineering The ‘stuffed’ Y-Matrix
I1 Y11 Y12 Y13 0 V1 I Y Y Y 0 V 2 = 21 22 23 2 I3 Y31 Y32 Y33 Y34 V3 I4 0 0 Y43 Y44 V4
Note that since no direct connections exist between Bus 1 & 4 or Bus 2 & 4, the corresponding entries in the Y matrix are zero 36 School of Engineering
Y-Matrix Math
37 School of Engineering PSC Reminder
For a passive network element(s), the PSC yields:
V1
V − V + I = 1 2 Z V I
- I = Y V1 − V2
38 V2 School of Engineering Y-Matrix Equations
Consider the following bus in a network:
Ia Bus 3
Z13 V3 V1 ZC1 Ib
Z 23 V Id I3 2 ZC2 Ic
39 School of Engineering Y-Matrix Equations KCL at bus 3 yields:
퐈ퟑ = 퐈퐛 + 퐈퐝 + 퐈퐚 + 퐈퐜
푽ퟑ 푽ퟑ 푽ퟑ − 푽ퟏ 푽ퟑ − 푽ퟐ 푰ퟑ = + + + 풁푪ퟏ 풁푪ퟐ 풁ퟏퟑ 풁ퟐퟑ
푽ퟑ − 푽풎 푰ퟑ = 푽ퟑ 풀ퟑ,풕풐풕풂풍 풕풐 품풏풅 + 풁풎ퟑ 풎 ≠ ퟑ 40 School of Engineering Y-Matrix Equations Algebra yields:
푽ퟑ − 푽풎 푰ퟑ = 푽ퟑ 풀ퟑ,풕풐풕풂풍 풕풐 품풏풅 + 풁풎ퟑ 풎 ≠ ퟑ
푰ퟑ = 푽ퟑ 풀ퟑ,풕풐풕풂풍 풕풐 품풏풅 + 풀풎ퟑ 푽ퟑ − 푽풎 풎 ≠ ퟑ
푰ퟑ = 푽ퟑ 풀ퟑ,풕풐풕풂풍 풕풐 품풏풅 + 풀풎ퟑ − 풀풎ퟑ 푽풎 41 풎 ≠ ퟑ 풎 ≠ ퟑ School of Engineering Y-Matrix Equations Finally:
푰ퟑ = 푽ퟑ 풀ퟑ,풕풐풕풂풍 풕풐 품풏풅 + 풀풎ퟑ − 풀풎ퟑ 푽풎 풎 ≠ ퟑ 풎 ≠ ퟑ
Or in general
푰풌 = 푽풌 풀풌,풕풐풕풂풍 풕풐 품풏풅 + 풀풎풌 − 풀풎풌 푽풎 풎 ≠풌 풎 ≠ 풌
42 School of Engineering Y-Matrix Equations Hence:
푰풌 = 푽풌 풀풌,풕풐풕풂풍 풕풐 품풏풅 + 풀풎풌 − 풀풎풌 푽풎 풎 ≠풌 풎 ≠ 풌
Admittance connected to the kth bus Admittance between the kth and mth bus
(i) (ii)
43 School of Engineering Y-Matrix Equations In terms of Y-matrix elements:
풀풌풌 = 풀풌,풕풐풕풂풍 풕풐 품풏풅 + 풀풎풌 (i) 풎 ≠풌
ퟏ 퐘퐤퐦 = 퐘퐦퐤 = − (ii) 퐙퐤퐦
44 School of Engineering Y-Matrix Equations Using the short-hand notation:
푰풌 = 푽풌 풀풌풌 + 푽풎 풀풎풌 풎 ≠ 풌
퐘퐤퐦 = 퐘퐦퐤
45 School of Engineering Consider a 3-Bus System
Bus 1 Bus 3
Z13
I1 I3
Bus 2
I2
46 School of Engineering
Let’s focus on Bus 2
퐈ퟐ = 퐕ퟐ 퐘ퟐퟐ + 퐕ퟏ 퐘ퟐퟏ + 퐕ퟑ 퐘ퟐퟑ
퐈ퟐ = 퐕ퟏ 퐘ퟐퟏ + 퐕ퟐ 퐘ퟐퟐ + 퐕ퟑ 퐘ퟐퟑ
푉1 퐼2 = 푌21 푌22 푌23 푉2 푉3
47 School of Engineering The General Case
For a n-Bus system we have:
퐼1 푌11 ⋯ 푌1푛 푉1 ⋮ = ⋮ ⋱ ⋮ ⋮ 퐼푛 푌푛1 ⋯ 푌푛푛 푉푛
Known sources The Y-Matrix To be calculated
48 School of Engineering
Power Flow Equations
49 School of Engineering Power Flow Equation
The current at bus k in an n-bus system is:
푰 풌 = 풀풌ퟏ푽 ퟏ + 풀풌ퟐ푽 ퟐ + ⋯ + 풀풌풏푽 풏
풏
푰 풌 = 풀풌풎푽 풎 풎=ퟏ
where
풋휽풌 푽 풌 = 푽풌풆 풋휽풎 푽 풎 = 푽풎풆 풋휹풌풎 풀풌풎 = 풀풌풎풆 50 School of Engineering Power Flow Equation
The complex power S at bus k is given by:
∗ 푺풌 = 푽풌푰풌 = 푷풌 + 풋푸풌
풏 ∗
푺풌 = 푽 풌 풀풌풎푽 풎 풎=ퟏ
51 School of Engineering Power Flow Equation
Substituting the voltage and admittance phasors into the complex power equations yields:
퐧 ∗
퐣훉퐤 퐣훅퐤퐦 퐣훉퐦 퐒퐤 = 퐏퐤 + 퐣퐐퐤 = 퐕퐤퐞 퐘퐤퐦퐞 퐕퐦퐞 퐦 = ퟏ
By Euler’s Theorem
∗ 푽풌 푽풎 = 푽풌 푽풎 퐜퐨퐬 휽풌풎 + 풋 퐬퐢퐧 휽풌풎 where 휽풌풎 = 휽풌 − 휽풎 52 School of Engineering Power Flow Equation
Expanding the complex power summation into real and imaginary parts and associating these with P & Q respectively yields our fundamental PFA equations
풏
푷풌 = 푽풌 푽풎 풀풌풎 퐜퐨퐬 휽풌 − 휽풎 + 휹풌풎 풎 = ퟏ
풏
푸풌 = 푽풌 푽풎 풀풌풎 퐬퐢퐧 휽풌 − 휽풎 + 휹풌풎 풎 = ퟏ 53 School of Engineering Power Flow Equation
Key SUMMARY points:
The bus voltage equations, I=YV, are linear
The power flow equations are quadratic, V2 appears in all P & Q terms
Consequently simultaneously controlling bus voltage and power flow is a non- linear problem
In general this forces us to use numerical solution methods for solving the power flow equations while controlling bus voltages 54 School of Engineering
Network Terminology
55 School of Engineering Simple 3-Bus Power System
PV Bus
Z12 ~ 1 2 ~
Slack Bus
3 PQ Bus
56 School of Engineering
Bus Types Total number of Quantities Quantities to be Bus Type buses Specified obtained Load Bus (or) n-m P, Q V , δ PQ Bus Generator Bus (or) Voltage Controlled m-1 P, V Q , δ Bus (or) PV Bus Slack Bus (or) Swing Bus (or) 1 V , δ P, Q Reference Bus 57 School of Engineering Bus Terminology
P = PG – PD
Q = QG – QD
PG & QG are the Real and Reactive Power supplied by the power system generators connected
PD & QD are respectively the Real and Reactive Power drawn by bus loads
V = Magnitude of Bus Voltage δ = Phase angle of Bus Voltage
58 School of Engineering
The Need for a Slack Bus
TL losses can be estimated only if the real and reactive power at all buses are known.
The powers in the buses will be known only after solving the load flow equations.
For these reasons, the real and reactive power of one of the generator bus is not specified.
This special bus is called Slack Bus and serves as the system reference and power balancing agent. 59 School of Engineering
The Need for a Slack Bus
It is assumed that the Slack Bus Generates the real and reactive power of the Transmission Line losses.
Hence for a Slack Bus, the magnitude and phase of bus voltage are specified and real and reactive powers are obtained through the load flow equation.
푺풖풎 풐풇 푪풐풎풑풍풆풙 푺풖풎 풐풇 푪풐풎풑풍풆풙 푻풐풕풂풍 푪풐풎풑풍풆풙 푷풐풘풆풓 = + 푷풐풘풆풓 풐풇 푮풆풏풆풓풂풕풐풓 푷풐풘풆풓 풐풇 푳풐풂풅풔 풍풐풔풔 풊풏 푻풓풂풏풔풎풊풔풔풊풐풏 푳풊풏풆풔 or
푻풐풕풂풍 푪풐풎풑풍풆풙 풑풐풘풆풓 푺풖풎 풐풇 푪풐풎풑풍풆풙 푺풖풎 풐풇 푪풐풎풑풍풆풙 = - 60 풍풐풔풔 풊풏 푻풓풂풏풔풎풊풔풔풊풐풏 푳풊풏풆풔 푷풐풘풆풓 풐풇 푳풐풂풅풔 푷풐풘풆풓 풐풇 푮풆풏풆풓풂풕풐풓 School of Engineering
Solving the PF equations
61 School of Engineering
Power Flow Analysis - Intro There are several methods available for solving the power flow equations:
Direct solutions to linear algebraic equations • Gauss Elimination
Iterative solutions to linear algebraic equations • Jacobi method • Gauss-Siedel method
Iterative solutions to nonlinear algebraic equations 62 • Newton-Raphson method School of Engineering
Gauss-Seidel Method
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a linear system of equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823.
63 School of Engineering The Jacobi Method
Carl Gustav Jacob Jacobi was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.
64 School of Engineering
Newton-Raphson Method
65 School of Engineering
Newton-Raphson Method
In numerical analysis, Newton's method (also known as the Newton–Raphson method), is named after Isaac Newton and Joseph Raphson. It is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
퐱; 퐟 퐱 = ퟎ
The Newton–Raphson method in one variable is implemented as follows: The method starts with a function f defined over the real numbers x The function's derivative f',
An initial guess x0 for a root of the function f.
If the function is ‘well behaved’ then a better approximation to the actual root is x1:
퐟(풙ퟎ) 풙ퟏ = 풙ퟎ − 66 퐟′(풙ퟎ) School of Engineering
Newton-Raphson Method
The Newton-Raphson method is a very popular non- linear equation solver as:
NR is very powerful and typically converges very quickly
Almost all PFA SW uses Newton-Raphson
Very forgiving when errors are made in one iterations; the final solution will be very close to the actual root 67 School of Engineering
Newton-Raphson Method
Procedure: Step 1: Assume a vector of initial guess x(0) and set iteration counter k = 0 (k) (k) (k) Step 2: Compute f1(x ), f2(x ),⋯ ⋯ fn(x )
Step 3: Compute ∆푚1, ∆푚2, ⋯ ⋯ Δ푚푛.
Step 4: Compute error = max[ ∆푚1 , ∆푚2 , ⋯ ⋯ ∆푚푛 ] Step 5: If error ≤ ∈ (pre-specified tolerance), then the final solution vector is x(k) and print the results. Otherwise go to step 6 Step 6: Form the Jacobin matrix analytically and evaluate it at x = x(k) T Step 7: Calculate the correction vector ∆푥 = [∆푚1, ∆푚2, ⋯ ⋯ Δ푚푛] by using equation d Step 8: Update the solution vector x(k+1) = x(k) + ∆푥 and update k=k+1 and go back to step 2. 68 School of Engineering
Newton-Raphson Method
Consider a 2-D example where in general there are n equations are given for the n unknown quantities x1, x2, x3 ⋯ xn.
f1(x1, x2) = k1 (Equation set a) f2(x1, x2) = k2
k1 and k2 are constants
Let’s assume that (0) (0) x1 ~ [x1 + ∆x1 ] (0) (0) x2 ~ [x2 + ∆x2 ]
Then (0) (0) (0) (0) k1 = f1(x1 + ∆x1 , x2 + ∆x2 ) 69 (0) (0) (0) (0) k2 = f1(x1 + ∆x1 , x2 + ∆x2 ) School of Engineering
Newton-Raphson Method
Applying the first order Taylor Expansion to f1 and f2 흏풇 흏풇 k ≅ f (x (0), x (0)) +∆x (0) ퟏ + ∆x (0) ퟏ 1 1 1 2 1 흏풙 2 흏풙 ퟏ ퟐ (Equation set b) (0) (0) (0) 흏풇ퟐ (0) 흏풇ퟐ k2 ≅ f2(x1 , x2 ) +∆x1 + ∆x2 흏풙ퟏ 흏풙ퟐ
‘Equation set b’ expressed in matrix form, 흏풇ퟏ 흏풇ퟏ k − f (x (0), x (0)) ∆x (0) 1 1 1 2 흏풙ퟏ 흏풙ퟐ 1 (Equation c) (0) (0) = (0) k2 − f2(x1 , x2 ) 흏풇ퟐ 흏풇ퟐ ∆x2
흏풙ퟏ 흏풙ퟐ
The Baby Jacobian 70 Matrix School of Engineering
Newton-Raphson Method
Then
(0) (0) ∆k1 (0) ∆x1 (0) = J (0) ∆k2 ∆x2
Adjusting the guess according to
(1) (0) (0) x1 = [x1 + ∆x1 ] (1) (0) (0) x2 = [x2 + ∆x2 ]
(N) Iterate until ∆xi < ∈ 71 School of Engineering
Newton-Raphson Method
For n equations in n unknown quantities x1, x2, x3 ⋯ xn.
f1(x1, x2 ⋯ ⋯ xn) = b1
f2(x1, x2 ⋯ ⋯ xn) = b2 (Equation set a) ⋮ fn(x1, x2, ⋯ ⋯ xn) = bn
72 School of Engineering
Newton-Raphson Method
In ‘equation set a’ the quantities b1, b2, ⋯ bn as well as the functions f1, f2 ⋯ ⋯fn are known.
To solve the equation set an initial guess is made; the initial guesses be denoted (0) (0) (0) as x1 , x2 , ⋯ xn .
First order Taylor’s series expansions (neglecting the higher order terms) are carried out for these equation about the initial guess
(0) (0) (0) (0) T The vector of initial guesses is denoted as x = [x1 , x2 , ⋯ xn ] .
Applying Taylor’s expansion to the “equation set a” will give: 73 School of Engineering
(0) (0) (0) 흏풇ퟏ 흏풇ퟏ 흏풇ퟏ f1(x1 , x2 ⋯ ⋯ xn ) + ∆풙ퟏ + ∆풙ퟐ + ⋯ + ∆풙풏 = b1 흏풙ퟏ 흏풙ퟐ 흏풙풏 흏풇 흏풇 흏풇 f (x (0), x (0) ⋯ ⋯ x (0)) + ퟐ ∆풙 + ퟐ ∆풙 + ⋯ + ퟐ ∆풙풏 = b 2 1 2 n 흏풙 ퟏ 흏풙 ퟐ 흏풙풏 2 ퟏ ퟐ (Equation set b) ⋮ (0) (0) (0) 흏풇풏 흏풇풏 흏풇ퟐ fn(x1 , x2 ⋯ ⋯ xn ) + ∆풙ퟏ + ∆풙ퟐ + ⋯ + ∆풙풏 = bn 흏풙ퟏ 흏풙ퟐ 흏풙풏
‘Equation set b’ can be written in matrix form as:
흏풇 흏풇 흏풇 ퟎ ퟏ ퟏ ⋯ ퟏ 흏풙 흏풙 흏풙풏 풇ퟏ(풙 ) ퟏ ퟐ ∆풙ퟏ 풃ퟏ ퟎ 흏풇 흏풇 흏풇 풇 (풙 ) ퟐ ퟐ ퟐ ퟐ ⋯ ∆풙ퟐ 풃ퟐ (Equation c) = 흏풙ퟏ 흏풙ퟐ 흏풙풏 = ⋮ ퟎ ⋮ ⋮ ⋮ 풇 (풙 ) 흏풇 흏풇 흏풇 ∆풙풏 풃 풏 풏 풏 ⋯ 풏 풏 흏풙 흏풙 흏풙풏 ퟏ ퟐ 74 School of Engineering
흏풇 흏풇 흏풇 ퟏ ퟏ ⋯ ퟏ 흏풙ퟏ 흏풙ퟐ 흏풙풏 흏풇 흏풇 흏풇 ퟐ ퟐ ⋯ ퟐ 흏풙ퟏ 흏풙ퟐ 흏풙풏 Jacobin Matrix (J) ⋮ 흏풇풏 흏풇풏 흏풇풏 ⋯ 흏풙ퟏ 흏풙풏 흏풙풏
Hence ‘equation c’ can be written as
ퟎ ∆풙 풃ퟏ − 풇ퟏ(풙 ) ∆풎 ퟏ ퟎ ퟏ ∆풙 풃 − 풇 (풙 ) ∆풎 ퟐ = [J-1] ퟐ ퟐ = [J-1] ퟐ (Equation d) ⋮ ⋮ ퟎ ⋮ ∆풙풏 풃풏 − 풇풏(풙 ) ∆풎풏
‘Equation d’ is the basic equation for solving the n algebraic equations given in ‘equation set a’. 75 NR Method Flowchart
76 School of Engineering
Power Flow Example (using Power World)
77 School of Engineering Power Flow Example
78 School of Engineering Power Flow Example, cont. Step 1: Calculate Y-Bus:
−푗10 푗10 0 YBus = 푗10 −푗30 푗20 0 푗20 −푗20
79 School of Engineering Power Flow Example, cont. Step 2: Define Bus types and their quantities:
Quantities Quantities to be Bus No Bus Type Specified obtained
One Slack Bus 푉1 , δ1 ---
Generator Bus Two 푃 , 푉 δ , 푄 PV 2 2 2 2
Load Bus Three 푃 , 푄 푉 , δ PQ 3 3 3 3
80 School of Engineering Power Flow Example, cont. Step 3: User Power Equations to solve for unknowns:
ퟑ
푷ퟐ = 푽ퟐ 푽풎 풀ퟐ풎 퐜퐨퐬 휽ퟐ풌 − 휹ퟐ + 휹풎 풎 = ퟏ
ퟑ
푷ퟑ = 푽ퟑ 푽풎 풀ퟑ풎 퐜퐨퐬 휽ퟑ풌 − 휹ퟑ + 휹풎 풎 = ퟏ
ퟑ
푸ퟑ = − 푽ퟑ 푽풎 풀ퟑ풎 퐬퐢퐧 휽ퟑ풌 − 휹ퟑ + 휹풎 풎 = ퟏ
81 School of Engineering Power Flow Example, cont.
ퟏ = ퟏퟎ 풔풊풏휹ퟐ − ퟐퟎ 푽ퟑ 풔풊풏(−휹ퟐ+휹ퟑ) Eq.1
ퟐ = ퟐퟎ 푽ퟑ 풔풊풏(−휹ퟑ+휹ퟐ) Eq.2
ퟐ ퟎ. ퟓ = ퟐퟎ 푽ퟑ 풄풐풔 −휹ퟑ+휹ퟐ − ퟐퟎ 푽ퟑ Eq.3
82 School of Engineering Power Flow Example, cont. Step 4: Apply Newton-Raphson on Eq.2 and Eq.3 to find 푉3 and (−훿3+훿2) ; then on Eq.1 to find 훿2 followed by 훿3
푽ퟑ= ퟎ. ퟗퟔퟖퟗ pu
휹ퟐ = −ퟓ. ퟕퟑퟗ Deg
휹ퟑ = −ퟏퟏ. ퟔퟔퟒ Deg
83 School of Engineering Power Flow Example, cont. Step 5: Use 푽ퟑ, 휹ퟐ, and 휹ퟑ to find 푸ퟐ and 푸ퟏ:
ퟑ
푸ퟐ = − 푽ퟐ 푽풎 풀ퟐ풎 퐬퐢퐧 휽ퟐ풌 − 휹ퟐ + 휹풎 풎 = ퟏ
ퟑ
푸ퟏ = − 푽ퟏ 푽풎 풀ퟏ풎 퐬퐢퐧 휽ퟏ풌 − 휹ퟏ + 휹풎 풎 = ퟏ
푸ퟐ = ퟎ. ퟕퟕퟔ pu
84 푸ퟐ = ퟎ. ퟕퟕퟔ pu School of Engineering Power Flow Example, cont. Final Soliton
Bus No V 휹 P Q
표 One 1 0 1 0.05
표 Two 1 −5.7392 1 0.776
표 Three 0.9689 −11.664 −2 -0.5
85 School of Engineering Power World Example
86 School of Engineering Power World Example, cont. Y-Bus
87 School of Engineering Power World Example, cont. Jacobin Matrix
88 School of Engineering Power World Example, cont. Power Flow Soliton
89