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: Grading Policy: Attendance (5%) Project (20%) 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%)

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 1 (---@uwindsor.ca) Student 2 (---@uwindsor.ca) Student 3 (---@uwindsor.ca) Project Title: Group 2: Student 1 (---@uwindsor.ca) Student 2 (---@uwindsor.ca) Student 3 (---@uwindsor.ca) Project Title: Group 3: Student 1 (---@uwindsor.ca) Student 2 (---@uwindsor.ca) Student 3 (---@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 Turbine Hydro (water) D » 10 m Non-uniform air-gap N S d-axis q-axis Hydrogenerator

Salient-Pole Synchronous Generator Stator Salient-pole rotor

Cylindrical-Rotor Synchronous Generator Stator Cylindrical rotor

Damper Windings

The rotor of the generator is driven by a prime-mover 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

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

Direct & Quadrature Axes d-axis q-axis Stator winding N Uniform air-gap Stator Rotor winding Rotor S Turbogenerator

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. [VA]base and [V]base are chosen first.

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 jXs jXl Ra + Ia + E d Vt 0o Equivalent circuit of a cylindrical-rotor synchronous machine

Phasor Diagram q-axis E IaXs d Vt IaXl f IaRa Ia d-axis

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 Lad=1.66 Laq=1.61 Ll=0.15 Ra=0.003 When the generator is delivering rated MVA at 0.9 p. f. (lag) and rated terminal voltage, compute the following: (i) Internal angle δi 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

(b) Compute the internal angle δi and field current ifd using the following equivalent circuit

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

d- and q-Axis Equivalent Circuits Ifd Xfd Rfd Xl pyd Ikd1 Imd Vtd Ra Id Xkd1 Xmd Rkd1 yq d-axis vfd + - pyfd pykd1 - yd pyq Ikq1 Imq Vtq Iq Xkq1 Xmq Rkq1 q-axis pykq1 Imd=-Id+Ifd+Ikd1 Imq=-Iq+Ikq1

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

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

The Flux Equations

Rearranged Flux Linkage equations

The Voltage Equations ……………..(1)

The Mechanical Equations where ……………..(2)

Linearized Form of the Machine Model ……………..(3)

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)

Substituting ∆Vtd and ∆Vtq in the flux equations: ……..(6)

Rearranging the flux equations in a matrix form: ………………...…..(7) where,

and…

Flux Linkage Equations (from the d- and q-axis equivalent circuits) Linearized flux linkage equations:

and thus, ………………………………………...(8)

: from (8) : inserting (8) into (7) where, : system state matrix ………..(9) : system state matrix

System to be Studied Infinite Bus Generator Vt It

System State Matrix and Eigen Values

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

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

Salient-pole synchronous generator 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