Advanced Power Systems

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Advanced Power Systems
Dr. Kar Sept. 16, 2008, Windsor
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• Dr. Kar 23 Old Drama Tel [phone]
  (ext.4796) Email nkar_at_uwindsor.ca Office Hour
  Thursday, 1200-200 pm
• http//www.uwindsor.ca/users/n/nkar/88-514.nsf
• Mariam Khan
• B20 Essex Hall Tel [phone] (ext.4792) Email
  khan11z_at_uwindsor.ca Office Hour Tuesdays,
  1200-200 pm

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• Course Text Book
• Electric Machinery Fundamentals by Stephen J.
  Chapman, 4th Edition, McGraw-Hill, 2005
• Electric Motor Drives Modeling, Analysis and
  Control by R. Krishnan Pren. Hall Inc., NJ, 2001
• Power Electronics Converters, Applications and
  Design by N. Mohan, J. Wiley Son Inc., NJ, 2003
• Power System Stability and Control by P. Kundur,
  McGraw Hill Inc., 1993
• Research papers
• Grading Policy
• Attendance (5)
• Project (20)
• Midterm Exam (30)
• Final Exam (45)

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Course Content

• Working principles, construction, mathematical
  modeling, operating characteristics and control
  techniques for synchronous machines
• Working principles, construction, mathematical
  modeling, operating characteristics and control
  techniques for induction motors
• Introduction to power switching devices
• Rectifiers and inverters
• Variable frequency PWM-VSI drives for induction
  motors
• Control of High Voltage Direct Current (HVDC)
  systems

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Exam Dates

• Midterm Exam Oct 28th 2008
• Final Exam Dec 9th 2008.

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• Term Projects
• Group 1Student 1 (---_at_uwindsor.ca)Student 2
  (---_at_uwindsor.ca)Student 3 (---_at_uwindsor.ca)
• Project Title Group 2Student 1
  (---_at_uwindsor.ca)Student 2 (---_at_uwindsor.ca)Stud
  ent 3 (---_at_uwindsor.ca)Project Title Group
  3Student 1 (---_at_uwindsor.ca)Student 2
  (---_at_uwindsor.ca)Student 3 (---_at_uwindsor.ca)

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Synchronous Machines

• Construction
• Working principles
• Mathematical modeling
• Operating characteristics

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• CONSTRUCTION

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• Salient-Pole Synchronous Generator

• Most hydraulic turbines have to turn at low
  speeds (between 50 and
  300 r/min)
• A large number of poles are required on the rotor

Hydrogenerator
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Salient-Pole Synchronous Generator

• Stator

• Salient-pole rotor

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Cylindrical-Rotor Synchronous Generator

• Stator

• Cylindrical rotor

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Damper Windings
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Operation Principle

• The rotor of the generator is driven by a
  prime-mover
• A dc current is flowing in the rotor winding
  which produces a rotating magnetic field within
  the machine
• The rotating magnetic field induces a
  three-phase voltage in the stator winding of the
  generator

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Electrical Frequency
Electrical frequency produced is locked or
synchronized to the mechanical speed of rotation
of a synchronous generator  
  where fe electrical frequency in Hz P
number of poles nm mechanical speed of the
rotor, in r/min
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Direct Quadrature Axes
Stator winding
N
Uniform air-gap
Stator
Rotor winding
Rotor
S
Turbogenerator
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PU System
Per unit system, a system of dimensionless
parameters, is used for computational convenience
and for readily comparing the performance of a
set of transformers or a set of electrical
machines.
Where actual quantity is a value in volts,
amperes, ohms, etc. VAbase and Vbase are
chosen first.
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Classical Model of Synchronous Generator

• The leakage reactance of the armature coils, Xl
• Armature reaction or synchronous reactance, Xs
• The resistance of the armature coils, Ra
• If saliency is neglected, Xd Xq Xs

jXl
Ra
jXs

Ia

Vt 0o
E d
Equivalent circuit of a cylindrical-rotor
synchronous machine
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Phasor Diagram
q-axis
E
IaXs
d
Vt
IaXl
f
IaRa
Ia
d-axis
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• The following are the parameters in per unit on
  machine rating of a 555 MVA, 24 kV, 0.9 p.f., 60
  Hz, 3600 RPM generator
• Lad1.66 Laq1.61 Ll0.15 Ra0.003
• When the generator is delivering rated MVA at 0.9
  p. f. (lag) and rated terminal voltage, compute
  the following
• (i) Internal angle di in electrical degrees
• (ii) Per unit values of ed, eq, id, iq, ifd
• (iii) Air-gap torque Te in per unit and in
  Newton-meters

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(b) Compute the internal angle di and field
current ifd using the following
equivalent circuit
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Direct and Quadrature Axes

• The direct (d) axis is centered magnetically in
  the center of the north pole
• The quadrature axis (q) axis is 90o ahead of the
  d-axis
• q angle between the d-axis and the axis of phase
  a
• Machine parameters in abc can then be converted
  into d/q frame using q
• Mathematical equations for synchronous machines
  can be obtained from the d- and q-axis equivalent
  circuits
• Advantage machine parameters vary with rotor
  position w.r.t. stator, q, thus making analysis
  harder in the abc axis frame. Whereas, in the d/q
  reference frame, parameters are constant with
  time or q.
• Disadvantage only balanced systems can be
  analyzed using d/q-axis system

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d- and q-Axis Equivalent Circuits
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Small disturbances in a power system

• Gradual changes in loads
• Manual or automatic changes of excitation
• Irregularities in prime-mover input, etc.

Importance of steady-state stability

• Knowledge of steady-state stability provides
  valuable information about the dynamic
  characteristics of different power system
  components and assists in their design
• - Power system planning
• - Power system operation
• - Post-disturbance analysis

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Related Terms

• Generator Modeling using the d- and q-axis
  equivalent circuits
• Transmission System Modeling with a RL circuit
• A Small Disturbance is a disturbance for which
  the set of equations describing the power system
  may be linearized for the purpose of analysis
• Steady-State Stability is the ability to maintain
  synchronism when the system is subjected to small
  disturbances
• Loss of synchronism is the usual symptom of loss
  of stability
• Infinite Bus is a system with constant voltage
  and constant frequency, which is the rest of the
  power system
• Eigen values and eigen vectors are used to
  identify system steady-state stability condition

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The Flux Equations
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Rearranged Flux Linkage equations
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The Voltage Equations
..(1)
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The Mechanical Equations
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Linearized Form of the Machine Model
..(3)
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Terminal Voltage
The d- and q-axis components of the machine
terminal voltage can be described by the
following equations
.(4)
where, Vt is the machine terminal voltage in per
unit. The linearized form of Vtd and Vtq are
.(5)
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Substituting ?Vtd and ?Vtq in the flux equations
..(6)
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Rearranging the flux equations in a matrix form
.....(7)
where,
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and
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Flux Linkage Equations (from the d- and q-axis
equivalent circuits)
Linearized flux linkage equations
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and thus,
...(8)
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where,
from (8)
inserting (8) into (7)
..(9)
system state matrix
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System to be Studied
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System State Matrix and Eigen Values
System State Matrix
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Eigen Values

• Eigen values are the roots of the characteristic
  equation
• Number of eigen values is equal to the order of
  the characteristic equation or number of state
  variables
• Eigen values describe the system response (
  ) to any disturbance

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Analyzing the Eigen Values of the System State
Matrix

• Compute the eigen values of the system state
  matrix, A
• The eigen values will give necessary information
  about the steady-state stability of the system
• Stable System If the real parts of ALL the eigen
  values are negative
• Example
• A system with the above eigen values is on the
  verge of instability

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Machine Parameters
Salient-pole synchronous generator 3kVA, 220V,
4-pole, 60 Hz and 1800 r/min