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Energy Engineering - Electric Power Systems

Full exam

Electric Power Systems Exam # 3: 04 /0 9/201 9 1° Part Exercise 1 A three -phase, four -pole, 50 Hz induction motor is supplied by a line -to-line voltage of 400 V. The windings of the induction motor are D -connected and the transformer ratio between the stator and the rotor is ������ = 1.2. In normal operating conditions the mo tor develops at its shaft a mechanical torque of ������������ = 150 �/������������������ , while the rotor angular speed is 1440 rot/min and the current in the rotor windings is �2= 25 ������. In no -load operating conditions the motor absorbs an apparent power ������0= 3000 ������������ with a power factor of cos ������0= 0.2. Neglecting the voltage drop in the stator windings, for the normal operating conditions, compute: 1. the mechanical power at the shaft of the motor ; 2. the Joule losses in the rotor; 3. the phase resistance and reactance of th e rotor windings; 4. the phasor of the current absorbed by the rotor from the supply network; 5. the efficiency of the motor; The reactive power required by the induction machine is provided by a synchronous generator ( SG ) while the real power is provided by the external grid ( Ext Grid , see the Figure below). The synchronous machine is characterized by synchronous reactance of 1.2 Ω and its stator phases are Y -connected. 6. compute the emf ������0̅̅̅ induced in the stator windings of the synchronous generator ; 7. the line current �������������̅̅̅̅ supplied by the external grid to the plant. Exercise 2 Consider the following electric network Line L [km] xl [/km] Vn [kV] L 10 0.4 20 Transformer An kT usc [MVA] [kV/kV] [% ] T 40 20/150 10 Generator Vn Real power output: P G Operating voltage: V G [kV] [MW] [kV] G 20 30 20.8 Demand Vn Absorbed real power: P D Power factor [kV] [MW] D 20 12. 5 0.9 5 - lagging Knowing that the External Grid (Ext Grid) is modeled as an infinite power generator that imposes at its terminals a voltage of 14 5.5 kV: 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 , compute the current phasors in the two branches of the electric gri d. 2° Part Answer only one of the two following questions 1. Starting from the general expression of the mmf of a rotating magnetic field at the air -gap of a two coaxial cylinder magnetic structure, i.e. ������(������,������)= ������������������������ cos (������ ∙������− ������+ ������0), define and explain the magnetic flux of a pole characterizing the rotating magnetic field, the flux linkage and the emf induced by the rotating magnetic field in a full -pitch winding installed at the air -gap and rotating with constant angular velocity ������������. 2. Starting from the structure of the induction (asynchronous) machine , obtain the equivalent circuit of the machine operated as a motor . In doing so, describe the operating conditio ns of the machine that lead to the definition of the equivalent circuit: hypothesis, physical phenomena, phasor diagrams. Simplifying hypothesis : the magnetic material is linear. Answer only one of the two following questions 1. The Power Flow system of equations : (a) derive of the equations; (b) explain the unknowns; (c) define and explain the bus types; (d) explain the balance of equations . 2. Describe the characteristic of the Jacobian matrix for transmission networks and comment its impact on the network operation. Then, based on the previously defined hypothesis derive the DC method for the power flow problem . In doing so, justify the introduced hypothesis. Equation sheet During the written test the candidate may consult this sheet, which will be delivered in conjunction with the test text. You are not allowed to consult any other material, such as your own notes, handouts or textbooks In the following equations the subscript m stands for the magnitude (absolute value) of the respective qua ntity. Power Flow Equations ��= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) � ��= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) � Newton -Raphson Method �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= ������∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −�������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −������∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= −�������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠�