A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery A lecture for electric power machinery
Advanced Power Systems Dr Kar U of Windsor Dr Kar 271 Essex Hall Email: nkar@uwindsor.ca Office Hour: Thursday, 12:00-2:00 pm http://www.uwindsor.ca/users/n/nkar/88-514.nsf GA: TBA B20 Essex Hall Email: TBA & TBA Office Hour: - 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 Project Midterm Exam Final Exam (5%) (20%) (30%) (45%) 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 Exam Dates Midterm Exam: Final Exam: Term Projects Group 1: Student ( -@uwindsor.ca) Student ( -@uwindsor.ca) Student ( -@uwindsor.ca) Project Title: Group 2: Student ( -@uwindsor.ca) Student ( -@uwindsor.ca) Student ( -@uwindsor.ca) Project Title: Group 3: Student ( -@uwindsor.ca) Student ( -@uwindsor.ca) Student ( -@uwindsor.ca) Synchronous Machines Construction Working principles Mathematical modeling Operating characteristics CONSTRUCTION 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 d-axis Nonuniform airgap N D ≈ 10 m q-axis Turbin e Hydro (water) Hydrogenerator S S N Salient-Pole Synchronous Generator Stator otor r e l o t-p Salien The Voltage Equations p (ψ d ) = vtd + Ra id +ψ q ω0 p (ψ kd1 ) = − Rkd1 ikd1 ω0 ( ) p ψ fd = v fd − R fd i fd ω0 ( ) p ψ q = vtq + Ra iq − ψ d ω0 ( ) p ψ kq1 = − Rkq1 ikq1 ω0 …………… (1) The Mechanical Equations dδ = ω − ω0 dt dω ω [Tm − Te ] = dt H where Te = ψ d I q −ψ q I d …………… (2) Linearized Form of the Machine Model ψ q0 • ∆ψ d = ∆vtd + Ra ∆id + ∆ψ q + ∆ω ω0 ω0 • ∆ψ kd = − Rkd ∆ikd ω0 • ∆ψ ω0 fd = ∆v fd − R fd ∆i fd • ψ ∆ψ q = ∆vtq + Ra ∆iq − ∆ψ d − d ∆ω ω0 ω0 • ∆ψ kq1 = − Rkq1 ∆ikq1 ω0 • ∆ δ = ∆ω ω ∆ω = [ ∆Tm − ∆Te ] 2H ∆Te = ψ d ∆I q + I q ∆ψ d −ψ q ∆I d − I d ∆ψ q …………… (3) Terminal Voltage The d- and q-axis components of the machine terminal voltage can be described by the following equations: vtd = Vt sin δ vtq = Vt cos δ ………………………….(4) where, Vt is the machine terminal voltage in per unit The linearized form of Vtd and Vtq are: ∆vtd = Vt cos δ ∗ ∆δ ∆vtq = −Vt sin δ ∗ ∆δ ……………………….…(5) Substituting ∆Vtd and ∆Vtq in the flux equations: ψ q0 • ∆ψ d = Vt cos δ • ∆δ + Ra ∆id + ∆ψ q + ∆ω ω0 ω0 • ∆ψ kd = − Rkd ∆ikd ω0 • ∆ψ ω0 fd = ∆v fd − R fd ∆i fd • ψ ∆ψ q = −Vt sin δ • ∆δ + Ra ∆iq − ∆ψ d − d ∆ω ω0 ω0 • ∆ψ kq1 = − Rkq1 ∆ikq1 ω0 • ∆ δ = ∆ω ω0 [ ∆Tm − ∆Te ] 2H ∆Te = ψ d ∆I q + I q ∆ψ d −ψ q ∆I d − I d ∆ψ q ∆ω = …… (6) Rearranging the flux equations in a matrix form: • ……………… … (7) ∆ X = [ S ][ ∆X ] + [ R ][ ∆I ] + [ B ][ ∆U ] where, • ∆ψ d • kd ∆ψ ∆•ψ fd • • ∆ X = ∆ψ q • ∆ψ kq1 • ∆δ • ∆ ω ∆ψ d ∆ψ kd ∆ψ fd ∆ X = ∆ψ q ∆ψ kq1 ∆ δ ∆ω ∆Id ∆ I kd1 ∆ I = ∆I fd ∆ I q ∆ I kq1 ∆v fd [ ∆U ] = ∆Tm and… 0 0 0 [ S ] = − ω0 0 0 − ω0 I q 2H − ω0 R fd [ R] = 0 0 ω0 0 0 ω0 I d 2H 0 0 0 ω0Vt cos δ 0 − ω0Vt sin δ 0 0 0 0 ω0 Ra 0 0 − ω0 Rkd 0 0 ω0ψ q 0 ω0 Ra 0 − ω0ψ d 2H 2H 0 ψ q0 −ψ d − ω0 Rkq1 0 ω0 0 [ B] = 0 0 ω0 2H Flux Linkage Equations (from the d- and q-axis equivalent circuits) ψ d − ( X md + X l ) ψ − X md kd ψ fd = − X md ψ q ψ kq1 X md X md X md X md ( X md + X fd ) 0 0 − ( X mq + X l ) 0 − X mq ( X md + X kd ) id i kd i fd X mq iq − ( X mq + X kq1 ) ikq1 Linearized flux linkage equations: ∆ψ d − ( X md + X l ) ∆ψ − X md kd ∆ψ fd = − X md ∆ ψ q ∆ψ kq1 X md ( X md + X kd1 ) X md 0 X md X md ( X md + X fd ) 0 0 − ( X mq + X l ) − X mq ∆id ∆i kd ∆i fd X mq ∆ i q − ( X mq + X kq1 ) ∆ikq1 0 and thus, ∆id − ( X md + X l ) ∆i − X md kd1 ∆i fd = − X md ∆iq ∆ikq1 − ( X md + X l ) −X md = − X md X md ( X md + X kd1 ) X md X md ( X md + X fd ) 0 − ( X mq + X l ) 0 − X mq X md ( X md + X kd1 ) ∆ψ d ∆ψ kd ∆ψ fd −1 = [ X reac ] ∆ψ q ∆ψ kq1 ∆δ ∆ω X md X mq − ( X mq + X kq1 ) −1 ∆ψ d ∆ψ kd ∆ψ fd ∆ψ q ∆ψ kq1 X md 0 X md 0 0 0 X mq X md ( X md + X fd ) 0 − ( X mq + X l ) 0 − X mq − ( X mq + X kq1 ) ……………………………………… (8) ∆ψ d 0 ∆ψ kd 0 ∆ψ fd ∆ψ q 0 ∆ψ kq1 0 ∆δ ∆ω ∆ψ d ∆ψ ∆id kd ∆i ∆ψ fd kd [ ∆I ] = ∆i fd = [ X reac ] −1 ∆ψ q = [ X reac ] −1[ ∆X ] ∆ i ∆ψ kq1 q ∆ikq1 ∆ δ ∆ω : from (8) • ∆ X = [ S ][ ∆X ] + [ R ][ ∆I ] + [ B ][ ∆U ] = [ S ][ ∆X ] + [ R ][ X reac ] −1[ ∆X ] + [ B ][ ∆U ] [ ] = [ S ] + [ R ][ X reac ] −1 [ ∆X ] + [ B ][ ∆U ] : inserting (8) into (7) = [ A][ ∆X ] + [ B ][ ∆U ] where, [ A] = [[ S ] + [ R ][ X reac ] −1 ] ……… (9) : system state matrix System to be Studied Vt It Generator Infinite Bus System State Matrix and Eigen Values System State Matrix: [ A] = [[ S ] + [ R ][ X reac ] −1 ] Eigen Values: λ1 , λ2 = −σ ± jω jω λ1 θ λ2 σ Eigen Values o Eigen values are the roots of the characteristic equation • ∆ X = [ A][ ∆X ] + [ B ][ ∆U ] o o Number of eigen values is equal to the order of the characteristic equation or number of state variables λ1t e Eigen values describe the system response ( ) to any disturbance Analyzing the Eigen Values of the System State Matrix o o o 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: o λ1 , λ2 = −0.15 ± j 2.0 λ3 = −0.0005 A system with the above eigen values is on the verge of instability Machine Parameters Salient-pole synchronous generator 3kVA, 220V, 4-pole, 60 Hz and 1800 r/min Machine parameters Per unit values d-axis magnetizing reactance, Xmd 1.189 q-axis magnetizing reactance, Xmq 0.7164 Armature leakage reactance, Xl 0.100 Field circuit leakage reactance, Xfd 0.276 First d-axis damper circuit leakage reactance, Xkd1 0.181 First q-axis damper circuit leakage reactance, Xkq1 0.193 Armature winding resistance, Ra 0.0186 Field winding resistance, Rfd 0.0058 First d-axis damper winding resistance, Rkd1 0.062 First q-axis damper winding resistance, Rkq1 0.052 ... TBA & TBA Office Hour: - Course Text Book: Electric Machinery Fundamentals by Stephen J Chapman, 4th Edition, McGraw-Hill, 2005 Electric Motor Drives – Modeling, Analysis and Control... generator Electrical Frequency Electrical frequency produced is locked or synchronized to the mechanical speed of rotation of a synchronous generator: fe = nm P 120 where fe = electrical... computational convenience and for readily comparing the performance of a set of transformers or a set of electrical machines PU Value = Actual Quantity Base Quantity Where ‘actual quantity’ is a value