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Energy Engineering - Electric Power Systems

Full exam

Electric Power Systems Exam # 2: 23 /0 7/201 9 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 7 MVA Nominal operating voltage (line -to-line) 3 kV Phase synchronous reactance 1.3 Ω Phase resistance 0.2 Ω Maximum turbine mechanical power 6.5 MW emf ������0 (phase -to-ground) at maximum e xcitation current 3.3 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 equally loaded, the emf of both generators is ������0= 2.6 ������������ while the load angle is ������= 13 .5° . 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 the generators to the plant’s load; 3. considering that SG1 is producing the entire real po wer 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 capabilit y limits for both synchronous generators of the plant. 5. considering the operating conditions identified at point 3 and assuming the plant’s load is constant, by how much must SG2 reduce its reactive power production such that SG1 operates at power factor ������������������ �������= �.������ – lagging maintaining constant its real power output . Exercise 2 Consider the following electrical network Line L [km] xl [/km] Vn [kV] L1 .1 55 0.4 220 L1.2 55 0.4 220 L2 40 0.4 220 Generator G An Vn Real power output: P G Operating voltage: V G [MVA] [kV] [MW] [kV] G 600 220 400 22 9.9 Demand D Vn Absorbed apparent power: A D Power factor [kV] [MVA] [-] D 220 300 0.95 - leading Knowing that the External Grid is modeled as an infinite power generator which imposes a t its busbars a voltage of 215.6 kV, after careful examination of the network topology: 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. compute a first approximation for the phases of the bus voltages using the DC Power Flow model ; 5. updating the starting profile for the iterative process of PF with the results obtained at question 4 perform one PF iteration using the Newton -Rhapson method (compute the bus voltage phasors ); 6. compute, using the results of question 5, the reactive power supplied by generator G to the network; 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. In doing so, explain the physical phenomena (laws) represented in the mathematical model. Then, using the obtained mathematical model of the machine derive and explain the mechanica l characteristic of the mach ine operating as a motor. 2. Starting from the structure of a round synchronous machine , obtain the equivalent circuit of the synchronous machine operated as a generator . In doing so, describe the operating conditions 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. Using a small electric network as example, demonstrate the const ruction of the Bus Admittance matrix according to the theory of electric circuits: (i) define the involved physical quantities, vectors and matrices; (ii) define the KC and Ohm laws in matrix form and (iii) obtain the Bus Admittance matrix. 2. The Newton -Raph son Method for the solution of the Power Flow (PF) system of equations: ( i) define and explain the principle of the Newton -Raphson Method for a generic non -linear equation in one unknown; and (ii) show the application of the Newton -Raphson Method to the PF 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 mater ial, 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 �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= �∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙�������� ∙��������� ∙������������� (�������− �������− ��������) �������� �������������= −�������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −�∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= −�������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠�