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Energy Engineering - Electric Power Systems
Full exam
Electric Power Systems Exam # 2: 21 /0 7/20 20 1° Part Exercise 1 The nameplate data of a three -phase synchronous generator is: Nominal Power 200 MVA Nominal voltage (line -to-line) 10 kV Nominal frequency 50 Hz Synchronous reactance 1 Number of poles 8 - Maximum no -load emf, E 0 (phase -to-ground) 13.5 kV Winding connections Triangle (D) - In normal operating conditions the synchronous generator is supplying a load of 35 + 0.5 ∙N MVA with a lagging power factor of cos ������ = 0.92 ; the synchronous generator is operating at nominal voltage. Neglect the power losses in the machine and consider the te rminal voltage of the generator always constan t. In these conditions : 1. determine the phasor of the current supplied by the generator to the load ( ������̅), the emf ( ������0) and the power angle ( ������; 2. determine the mechanical torque at the shaft of the generator in the given operating conditions; 3. considering the emf ������0 constant and fixed to the value previously determined (constant excitation current) and considering an increase in the mechanical power at the shaft of the machine of 4 (four) times the previous value determine the phasor of the current supplied by the generator to the load ( ������̅), and the power angle ( ������); 4. determine the maximum reactive power ( ������max ) that the machine can produce when operat ed at the mechanical power indicated in the previous point . In the above data, N is the last number of the student’s Person Code/Codice Persona (e.g. if Person Code is 10 38148 6 then N = 6); Please substitute accordingly. Exercise 2 Consider the following electric network Line xl [p.u.] L1 0.2 L2 0.1 L3 0.05 + 0.01 ∙N Generator Real power output: P G Operating voltage: V G [p.u.] [p.u. ] G 12 1.08 Demand Absorbed apparent power: A D Power factor [p.u.] D1 5 0.9 2 - lagging D2 1 0.9 0 - leading Knowing that the External Grid (Ext Grid) is modeled as an infinite power generator that imposes at its terminals a voltage of 1.04 + 0.005 ∙N p.u. : 1. determine the bus admittance matrix; 2. identify the bus types for Power Flow (PF) computation; 3. choose the starting profile for the iterative process of PF; 4. perform one PF iteration using the Newton -Rhapson method (compute the bus voltage phasors); 5. using the results of the previous point and knowing that the long -term thermal limit of line L2 is ������������2������������ = 900 , compute the current in this line ( ������12̅̅̅̅) and check against the long -term thermal limit . NOTE : In the above all quantities are in p.u. with respect to ������ = 100 ������������ and ������������ = 400 ������ . NOTE : All calculations are to be made in p.u. with respect to the given base. NOTE : In the above N is the last number of the student’s Person Code/Codice Persona (e.g. if Person Code is 1038148 6 then N = 6); Please substitute accordingly. 1 2 t rid 1 2° Part Answer only one of the two following questions 1. Assuming a two poles rotating electric machine and a rotating magnetic field at the air gap with a magneto -motive force given by (������,������)= ������������������ cos (������ ∙������− ������+ ������0) obtain the emf induced by this rotating magnetic field into a full -pitch coil rotating with constant angular velocity ������������. It is reminded that the flux of the pole characterizing the rotating magnetic field is : ������ = ∫ ∙������ = 2 ������∙������������ ∙∙������ where ������������ is the maximum value of the ma gnetic induction at the air -gap of the rotating magnetic field; is the length of the machine and ������ is the length of the half -circumference of the air -gap defined by the zero crossing points of the rotating magnetic field. Answer the question by adopting the following path: a. define the flux linkage in the rotating winding and describe how it is calculated , also through the appropriate use of figures (choice of surface to integrate, choice of dA and identification of ������ and ������, etc. ); b. obtain the emf induced in the rotating winding . 2. Starting from the equivalent circuit of the synchronous machine while neglecting the losses in the machine, answer the following questions : a. Determine the complex power generated by the machine at its external terminals as a function of terminal voltage, induced emf in the stator windings and load angle; b. Based on the p reviously obtained results obtain the electrodynamic torque ( ������) characteristic of the machine and depict it graphically in the ������− Ω and ������− δ planes ; c. Considering the lowering of the excitation current as a disturbance in the machine , define and demon strate the stable operating range of the machine . Answer only one of the two following questions 1. The Power Flow system of equations : a. define and derive the equations; b. explain the unknowns; c. define and explain the bus types; d. explain the balance of equations. 2. The Newton -Raphson method : a. Prove the iterative formula and describe the method’s iterative process steps using the Power Flow equations (define and describe all the involved quantities) . b. Considering an electric network made of u PQ buses, g PV buses and one Slack bus provide the dimensions of the Jacobian matrix and of its four building blocks (submatrices) . c. Describe , while motivating , the choice of a suitable initial guess for the Newton -Raphson method when applied to solve the Power Flow equations . Equation sheet In the following equations the superscript m stands for the magnitude (absolute value) of the respective quantity. Power Flow Equations = ������ ∙∑ ������ ∙������ ∙������������ (������− ������− ������ ) = ������ ∙∑ ������ ∙������ ∙������������ (������− ������− ������ ) Newton -Raphson Method ������ ������������= ������ ∙������ ∙������������ (������− ������− ������ ) ������ ������������= ������∙������ ∙������ ∙������������ (������ )+ ∑ ������ ∙������ ∙������������ (������− ������− ������ ) ≠ ������ ������������= ������ ∙������ ∙������ ∙������������ (������− ������− ������ ) ������ ������������= −������ ∙∑ ������ ∙������ ∙������������ (������− ������− ������ ) ≠ ������ ������������= ������ ∙������ ∙������������ (������− ������− ������ ) ������ ������������= −������∙������ ∙������ ∙������������ (������ )+ ∑ ������ ∙������ ∙������������ (������− ������− ������ ) ≠ ������ ������������= −������ ∙������ ∙������ ∙������������ (������− ������− ������ ) ������ ������������= ������ ∙∑ ������ ∙������ ∙������������ (������− ������− ������ ) ≠