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Energy Engineering - Electric Power Systems
Full exam
Electric Power Systems Exam #1 : 30 /0 6/20 20 1° Part Exercise 1 The power plant PP depicted in the figure bellow consists of two identical generation groups (synchronous machine SG and transformer T) characterized by the following data: Transformer T: Nominal power 50 (kVA) Nominal transformer ratio 15/3.8 (kV/kV) Short -circuit voltage (v sc) 10 (%) Synchronous machine SG: Nominal power 45 (kVA) Nominal voltage (line -to-line) 3.8 (kV) Stator winding connection Star Synchronous reactance (x s) 45 ( ) Synchronous resistance (r s) 5 ( ) Maximum turbine mechanical power 40 (kW) Maximum electromotive force (phase) 2.7 (kV) The power plant PP operates at a line -to-line voltag e ������������������ = 14 .5+ 0.2∙������ (kV) and produces a total real power of ������������ = 60 + 2∙������ (kW) with a lagging power factor of cos ������������������ = 0.90 . Moreover, as depicted in the figure, the complex power produced by the power plant is evenly distributed between the two generation groups. For the given operating conditions: 1. determine the real and reactive powers produced by each SG (������1/������1 and ������2/������2); 2. determine the magnitudes of the line -to-line voltages at the terminals of the two SG , ������������1 and ������������2; 3. determine the phasors of the emf of the two SG (������������1 ̅̅̅̅̅ and ������������2 ̅̅̅̅̅); 4. draw, in a qualitative manner, the phasor diagrams of both SG ; 4. determine the mechanical power of the turbines of each SG ; 5. check the capability limits for both SG ; 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 0.1 L2.1 0.08 + 0.1 ∙N L2 .2 0.18 Generator Real power output: P G Operating voltage: V G [p.u. ] [kV] G 1.25 1.02 + 0.02 ∙N Demand Absorbed apparent power: A D Power factor [p.u. ] D1 3.5 + 0.1 ∙N 0.9 0 - lagging D2 0.5 0.9 5 - lagging NOTE : In the above tables all quantities are in p.u. with respect to ������������ = 100 ������������������ and ������������ = 150 ������������ ; In the above tables 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. Knowing that the External Grid (Ext Grid) is modeled as an infinite power generator that imposes at its terminals a voltage of 1.04 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 . NOTE : All calculations are to be made in p.u. with respect to the given base. t rid 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 c ircuit, 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 capa bility curve of a round synchronous machine : a. define the adopted simplifying hypothesis to construct the capability curve; b. define and explain the operational limits considered in the capability curve of a round synchronous machine; 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 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 ve ctors 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 DC Power Flow method: a. Describe what the method aim (goal) is and name the adopted simplifying hypothesis; b. Consider a generic branch connecting bus p to bus q and derive the nodal power flow equations for the generic bus p starting from the complex power flowing in this branch; in doing this, show the impa ct of the previously introduced simplifying hypotheses; c. Write the DC PF model in matrix form . 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 ������ ������������= ������ ∙������ ∙������������ (������− ������− ������ ) ������ ������������= ������∙������ ∙������ ∙������������ (������ )+ ∑ ������ ∙������ ∙������������ (������− ������− ������ ) ≠ ������ ������������= ������ ∙������ ∙������ ∙������������ (������− ������− ������ ) ������ ������������= −������ ∙∑ ������ ∙������ ∙������������ (������− ������− ������ ) ≠ ������ ������������= ������ ∙������ ∙������������ (������− ������− ������ ) ������ ������������= −������∙������ ∙������ ∙������������ (������ )+ ∑ ������ ∙������ ∙������������ (������− ������− ������ ) ≠ ������ ������������= −������ ∙������ ∙������ ∙������������ (������− ������− ������ ) ������ ������������= ������ ∙∑ ������ ∙������ ∙������������ (������− ������− ������ ) ≠