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Energy Engineering - Electric Power Systems

Full exam

Electric Power Systems Exam #1 : 28/0 6/20 22 1° Part Exercise 1 In the figure bellow a three -phase demand (D) is supplied by a power plant made of two synchronous generators, SG 1 and SG 2, respectively, through the electric line L. The demand is supplied at line -to -line voltage of ������������������������������������������ = 9.8 [kV] and absorbs �������= 4 [MW ] with a lagging power factor cos ������������= 0.92 . Initially, breaker B connecting SG2 to the network is opened so generator SG2 is offline. The electric line is characterized by �������= 0.5 Ω and �������= 1.5 Ω. The synchronous generator nameplate s specif y the following data: SG1 SG2 Nominal apparent power MVA 6 5 Nominal operating voltage (line -to -line) kV 10 10 Phase synchronous reactance Ω 20 15 Maximum turbine mechanical power MW 5.5 4.5 Emf ������������ (phase -to -ground) at maximum excitation current kV 11.5 6.65 Stator windings connection - Delta Y(Star) Determine: 1. the line -to -line voltage at the power plant terminals (busbars) and the real and reactive power supplied by the synchronous generator SG 1 to the grid; 2. the emf ������0 and the load angle ( ������) of the synchronous generator SG 1; 3. check the capability curve of the synchronous generator SG 1 in the given operating conditions; 4. determine the maximum reactive power ( ��������1max ) that the machine can produce when operated at the electrical power previously obtained . 5. considering breaker B closed and, thus, SG 2 in operation and in no -load conditions, find the minimum reactive power SG 2 nee ds to produce so that SG 1 capability curve is satisfied and the power plant works in the same operating conditions determined previously . Exercise 2 Consider the following electric network Line L [km] xl [/km] Vn [kV] L 10 0.4 20 Shunt Capacitor Xc [] Vn [kV] C 200 20 Transformer An kT usc [MVA] [kV/kV] [% ] T 5 20/150 12.5 Generator Vn Real power output: P G Operating voltage: V G [kV] [MW] [kV] G 20 4 19.4 Demand Vn Absorbed real power: P D Power factor [kV] [MW] D1 20 6 0.9 0 - lagging D2 20 1 0.95 – lagging Knowing that the External Grid (Ext Grid) is modeled as an infinite power generator that imposes at its terminals a voltage of 147 kV : 1. Calculate the bus admittance matrix; 2. Identify the bus types for Power Flow (PF) computation; 3. Select 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 first iteratio n, calculate the following: • The reactive power of the generator G - QG [MVAr] • The active power flow through the transformer T – P1-2 [MW] 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 subscri pt m stands for the magnitude (absolute value) of the respective quantity. Power Flow Equations ��= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) � ��= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) � Newton -Raphson Method �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= ������∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −�������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −������∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= −�������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� 2° Part Answer only one of the two following questions 1. The rotating magnetic field generated by a full -pitch coil supplied with a DC current and installed in the rotor of a rotating electric machine: a. Draw and describe the structure of the magnetic circuit and the adopted simplifying hypothesis; b. using the unfolded airgap of the magnetic circuit, obtain the expression of the fundamental component of the mmf at the air gap; c. starting from the expression of the fundamental component of the mmf at the airgap show how the expression of the mmf of the rotating magnetic field is obtained. 2. The capability curve of a round synchronous machine : a. define the adopted simplifying hypothesis to construct the capability curve regarding machine losses, voltage at machines terminals and their impact on the model of the machine; b. define and explain the physical operational limits considered in the capabilit y curve; c. show, while motivating, the procedure to graphically construct the capability curve starting from the voltage phasor -diagram of the machine. Answer only one of the two following questions 1. The bus admittance matrix : a. starting from the graph of a simple electric grid define the vectors and matrices necessary to build the bus admittance matrix according to the circuit theory method; Explain the quantities defined . b. prove, using the previously defined vectors and matrices, that [�̅]≜ [������]∙[������̅]∙[������]�; in doing so, also define and justify the main equations needed. c. describe the proprieties of the bus admittance matrix. 2. The Secondary Frequency regulation : a. Explain why, in gener al, the frequency regulation is required in the electric power systems and what is the goal of secondary frequency regulation . b. Explain how the Secondary Frequency regulation works in a multi -machine power system with the help of the linear f-P regulating characteristics of the machines .