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Energy Engineering - Electric Power Systems

Full exam

Electric Power Systems Exam # 4: 28 /01 /20 22 1° Part Exercise 1 Consider a generation plant made of two identical synchronous machines connected in parallel to the plant terminals as shown in the figure bellow . Each synchronous machine is characterized by: Nominal apparent power 16 MVA Nominal operating voltage (line -to-line) 3 kV Phase synchronous reactance 1.3 Ω Phase resistance 0.2 Ω Maximum turbine mechanical power 15 MW emf ������0 (phase -to-ground) at maximum excitation current 3.9 kV Stator windings connection Y In normal operating conditions the voltage at the plant terminals is equal to the nominal voltage of the synchronous machines, the two synchronous generators are equ ally loaded, the emf of both generators is ������0= 3.1 ������� while the load angle is ������= 25° . 1. compute the current phasor supplied by the generators to the plant’s load and the corresponding power factor; 2. compute the real and reactive powers supplied by th e generators to the plant’s load; 3. considering that SG1 is producing the entire real power required by the plant’s load while SG2 the entire reactive power required by the plant’s load, compute the emf phasor ������0̅̅̅ for each of the two synchronous generators of the plant; 4. for the operating conditions identified at point 3 check the capability limits for both synchronous generators of the plant. 5. considering the operating conditions identified at point 3 and assu ming the plant’s load is constant, by how much must SG2 change its reactive power production such that SG1 operates at power factor ������������������ �������= �.������ – lagging maintaining constant its real power output . Exercise 2 Consider the following electric network Line xl [pu ] L1 0.4 L2.1 0.3 L2 .2 0.35 Generator Real power output: P G Operating voltage: V G [pu ] [pu ] G 1 1.0 2 Demand Absorbed apparent power: A D Power factor [pu ] D 0.5 0.9 - leading Knowing that the capacitor bank C is characterized by a reactance ������C = 2 p.u. and that the External Grid ( Ext Grid ) is modeled as an infinite power generator that imposes at its terminals a voltage of 0.96 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 , compute the reactive power produced by generator G. 2° Part Answer only one of the two following questions 1. Starting from the structure of the Elementary Electric Machine obtain the mathematical model of the machine. To do so, answer the following points: a. explain the physical phenomena (laws) present in the machine. b. derive the mathematical model. c. then, using the obtained mathematical model of the machine derive and explain the mechanical characteristic of the machine operating as a motor. 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 symmetrical AC currents wil l 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. De scribe 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 ferromagnetic material and emf induced in a distributed turns stato r 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 machine using the Behn -Eschemburg method . Answer only one of the two following questions 1. The bus admittance matrix: a. starting from the graph representation 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. Considering the block structure of the Jacobian matrix of the Newton -Rhapson method applied to the power flow equations: ������= [ ������� ������������ ������� ������������ ������� ������������ ������� ������������ ] Note: bold denotes matrix Answer the following questions: a. Consider the case of a generic HV transmission network and enunciate the main simplifying hypothesis that can be adopted; b. Show the impact of the simplifying hypothesis on the structure of the Jacobian matrix; c. comment on the consequences on network operation. 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 �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= �∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −�������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −�∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= −�������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠�