logo
  • userLoginStatus

Welcome

Our website is made possible by displaying online advertisements to our visitors.
Please disable your ad blocker to continue.

Current View

Energy Engineering - Electric Power Systems

Full exam

Electric Power Systems Exam #1 : 29 /0 6/20 21 1° Part The nameplate data of a three -phase synchronous generator is : Nominal Power 180 MVA Maximum mechanical power of turbine 160 MW Nominal voltage (line -to-line) 10 kV Nominal frequency 50 Hz Synchronous reactance 1.5  Number of poles 4 - Maximum emf, E E (phase -to-ground) 12.5 kV Winding connections Triangle (D) - In normal operating conditions the synchronous generator is injecting a line current of 7000 + 250 ∙N A to the external grid with a leading power factor of cos ������ = 0.94 ; the synchronous generator is operating at its nominal voltage. The power losses in the machine are neglected . In these conditions : 1. determine the real (�������) and reactive power (�������) produced by the generator, the emf ( ������������) and the load angle (������); 2. draw the voltage phasor diagram corresponding the previously calculated operating conditions; 2. determine the mechanical torque at the shaft of the generator in the given operating conditions; 3. considering a change in the operating conditions of the generator described by an increase in the operating voltage at the external terminals of the generator of +1 ∙N % and an increase of the emf of +20 %, while the real power produced remains constant, determine the new value of the reactive power produced by the generator (�������), the magnitude of the line current injected by the generator to the external grid and the power factor; 4. draw the voltage phasor diagram corresponding the previously calculated operating conditions; 5. check the capability limits of the generator for the operating conditions described at question 3 . 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 .1 0.5 L1.2 0.4 + 0.05 ∙N L2 0.25 L3 0.2 Generator Real power output: P G Voltage set - point : V G [p.u ] [p.u ] G 1.5 1.04 Demand Absorbed real power: PD Power factor [p.u ] [-] D1 0.5 0.9 0 – leading D2 2.5 + 0.1 ∙N 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 1.06 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 -Raphson method (compute the bus voltage phasors); 5. using the results of question 4 , compute the reactive power generated by G. NOTE : In the above all quantities are in p.u. with respect to ��������� = 1000 ������������� 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. 1.1 t rid 1. 2° Part Answer only one of the two following questions 1. Assuming 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 : ������ = ∫ �∙������� = ������∙�������������� ∙�∙������ where �������������� is the maximum value of the magnetic induction at the air -gap of the rotating magneti c 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 such that the following questions are answered : a. define the flux linkage in the rotating winding and describe how it is calculated , also employing 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 structure of the synchronous generator, neglecting the real power losses of the machine and knowing that, according to Galileo Ferraris theorem, supplying the stator windings with a system of three balanced and symmet rical AC currents will result in the induction of an emf, ������������̅, in the stator windings given by ������������̅ = −�∙�∙������������̅ , where X is the synchronous reactance of the stator windings and ������������̅ is the phasor of the current in a generic stator winding , answer the following questions : a. Describe the no -load operating conditions of the synchronous generator: constructive elements of the machine, operating conditions, formation of the rotating magnetic field, linearity of the f erromagnetic material and emf induced in a distributed turn stator winding ; b. Describe the Load operating conditions of the synchronous generator: superposition of effects (and why ?) and phasor diagram of the machine ; c. Obtain the equivalent circuit of the ma chine using the Behn -Eschemburg method . Answer only one of the two following questions 1. The bus admittance matrix : a. starting from the graph of a simple grid define the vectors and matrices necessary to build the bus admittance matrix according to the circuit theory method; Explain the involved quantities. b. prove, using the previously defined vectors and matrices, that [�̅]≜ [�]∙[�̅]∙[�]������. c. describe the proprieties of the bus admittance matrix and the rules to calculate it using the network inspection method . 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 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 quantity. Power Flow Equations ��= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) � ��= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) � Newton -Raphson Method �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= ������∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −�������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −������∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= −�������� ∙�������� ∙��������� ∙������������� (�������− �������− ��������) �������� �������������= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠�