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Energy Engineering - Electric Power Systems

Full exam

Electric Power Systems Exam # 5: 20 /02 /20 20 1° Part Exercise 1 Consider the Elementary Electric Machine (EEM) illustrated in the Figure bellow: The EEM is supplied by a DC voltage V and the magnetic field B is constant in time. The characteristics of the EEM are: B d V R 2.0 T 100 cm 60 V 0.4 Ω The magnetic field B is perpendicular to the Figure’s plane and is exiting the Figure’s plane. Assuming the mobile iron bar, characterized by the electrical resistance R, can move along the fixed iron part without mechanical friction and at constant velocity: 1. draw, on the circuit of the EEM, the vectors of the mechanical and electrodynamic forces that act on the mobile part of the machine; 2. considering that at start -up, when the mobile part is still and breaker k1 is opened, determine the value of the start -up resistance ( Rsu) that will result in the reduction by half of the start -up electrodynamic force developed if breaker k1 would be closed ; 3. considering that in normal operating conditions breaker k1 is closed, determine the analytical expression of the mechanical characteristic of the machine ( �������� = ������(������)); 4. considering that breaker k1 is closed, that on the mobile iron bar a mechanical load is applied and that the characteristic of this load is gi ven by the equation ������������(������)= 90 + 0.15 ∙������ [������], determine the velocity of the mobile part ( v), the electrodynamic force acting on the mobile part ( �������� ), the emf induced by the magnetic field in the electric circuit ( E), the real power absorbed from the grid ( Pela) and the efficiency of the EEM ( ������); Exercise 2 Consider the following electrical network Line L [km] xl [/km] Vn [kV] L1 40 0.4 400 L2.1 50 0.4 400 L2.2 80 0.4 400 Generator G An Vn Real power output: P G Operating voltage: V G [MVA] [kV] [MW] [kV] G 600 220 100 416 Demand D Vn Absorbed apparent power: A D Power factor [kV] [MVA] [-] D 220 900 0.95 - leading Knowing that the inductive reactor I is characterized by a reactance ������������= 80 Ω and that the External Grid (Ext Grid ) is modeled as an infinite power generator which imposes a t its busbars a voltage of 428 kV, after careful examination of the network topology: 1. determine the bus admittance matrix; 2. identify the bus types for Power Flow (PF) c omputation; 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. determine the exact value of the power exchanged by the analyzed network with the External Grid ; 2° Part Answer only one of the two following questions 1. Enunciate and demonstrate the Galileo Ferraris theorem: (i) describe the structure of the electric machine; (ii) enunciate the hypothesis; (iii) demonstrate the existence of the rotating magnetic field and (iv) show the link between the position of the rotating magnetic field and the AC currents that generate it . 2. Starting from the equivalent circuit of the synchronou s machine describe the capability curves of the synchronous machine operated as a generator: definitions, construction starting from the voltage phasor diagram of the machine, highlights . Answer only one of the two following questions 1. Using a small electric network as example, demonstrate the construction of the Bus Admittance matrix according to the theory of electric circuits: (i) define the involved physical quantities, vectors and matrices; (ii) d efine the KC and Ohm laws in matrix form and (iii) obtain the Bus Admittance matrix . 2. The Newton -Raphson 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 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 �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= ������∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −�������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= �������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= −������∙�������� ∙��������� ∙������������� (�������� )+ ∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠� �������� �������������= −�������� ∙�������� ∙��������� ∙������������� (�������− �������− �������� ) �������� �������������= �������� ∙∑ �������� ∙��������� ∙������������� (�������− �������− �������� ) �≠�