ÈÖÓ º ĺ ÙÞÞ ÐÐ ÈÖÓ º ʺ ³ Ò Ö ½ ½¹¼ ¼¹¼¼ ËÝ Ø Ñ ÅÓ Ð Ò ÀË ¾¼½ µ Ü Ö ÌÓÔ ÀÝ ÖÓ Ð ØÖ ÔÓÛ Ö ÔÐ ÒØ À Èȵ ¹ È ÖØ ÁÁ Ð ÖÒ Ø Þº ÇØÓ Ö ½ ¾¼½ Problem (From the reservoir to the grid) The causality diagram of the entire system (i.e. including the second part of the HEPP exercise) is shown in Figure. Again, blocks with drop shadows are dynamic, the others are algebraic. This solution is not unique, for example you might chose to have volume or mass flows instead of velocities. The outputs of the dynamic blocks were chosen to be the level variables, algebraic blocks were introduced where required. The signals are explained in Table. Table : Signals in the causality diagram. Symbol h R h W p W p F v T v F v V V A V u x T G T T ω I U net p Description Water level of the reservoir Water level of the water tank Pressure after the water tank Pressure after the down pipe Velocity in the tunnel Velocity in the down pipe Velocity after the valve Dynamic volume of the compressibility Opening area of the valve Input of the valve s needle Position of the valve s needle Torque of the generator Torque of the turbine Angular velocity of turbine and generator Current Voltage of the grid Ambient pressure
Figure : The causality diagram of the hydroelectric power plant.
Turbine The turbine and the generator are connected through a rigid shaft. It can be assumed that the inertias of the turbine, generator and shaft are all summarized in the Θ value. The turbine s torque can be calculated using the momentum law: T T (t) = F T.5 d T urb = ρ d T urb (v V (t).5 d T urb ω(t) ) A V (t) v V (t). () where F T is the mean force acting on the turbine s blades. The rotational speed can be formulated using the principle of angular momentum dω(t) = Θ (T T (t) T G (t) ), (2) where the friction term d ω(t) was omitted, since much smaller compared to the other two torques. The generator s torque T G (t) is a function of the current I(t) and will be discussed in the next section. Table 2: Signals and parameters for the subsystem Turbine. Description Symbol Units Inputs Exit velocity after the valve v V m/s Opening area of the valve A V m 2 Torque of the generator T G N m Outputs Rotational speed of the turbine ω rad/s Parameters Density of the water ρ kg/m 3 Diameter of the turbine d T urb m Moment of inertia of turbine, generator and shaft Θ kg m 2 Generator und Grid L G R G L net R net + U ind (t) U net (t) I(t) Figure 2: Schematic of the circuit. The schematic of the LR-circuit is shown in Figure 2. The voltage induced in the generator is a function of the rotational speed: U ind (t) = κ ω(t). (3)
using the Kirchhoff s voltage law, the following equation can be written: κ ω(t) L tot di(t) R tot I(t) = () di(t) = κ ω(t) R tot I(t), L tot (5) where R tot = R G + R net and L tot = L G + L net. The generator s torque, which works against turbine s torque, is proportional to the current: T G (t) = κ I(t). (6) Therefore the grid s voltage can be calculated as following: U net (t) = L net di(t) + R net I(t). (7) Table 3: Signals and parameters for the subsystem Generator and Grid. Description Symbol Units Inputs Rotational speed of the turbine ω rad/s Outputs Generated current I A Torque of the generator T G Nm Voltage of the grid U net V Parameters Constant of the generator κ Vs Inductance of the generator L G Vs/A Resistance of the generator R G Ω Inductance of the grid L net Vs/A Resistance of the grid R net Ω
Problem 2 (Simulate the entire hydroelectric power plant) Once you will have connected all the subsystem similarly to the structure of the causality diagram, you should be ready to simulate the entire system. In case you were not able to do it, you can use the Simulink model provided with the solution. The equations contained in this model are the ones found in Exercise and 5. The initial condition for each integrator in the system are the following: x(t = ) = m h W (t = ) = 3 m v T (t = ) = m /s v F (t = ) = m /s V (t = ) = l F π d F 2 ω(t = ) = rad /s I(t = ) = A These are the system s equilibrium (i.e. steady state conditions) values when the valve is closed. In this case a solver that can handle stiff differential equations, like ode23s(stiff), was used. The trajectories of the system s state variables, for the given valve position s evolution, are shown in Figure 3. Valve s Position 25 Water tank s water level 7 x 6 Pressure before Valve.8 2 6 5 5 x [ ].6. h W [m] 5 p F [Pa] 3 2.2 395 39 5 Tunnel s Volume Flow 5 Downpipe s Volume Flow 25 Turbine s Rotational Speed 2 3 3 5 V T [m 3 /s] 2 V F [m 3 /s] 2 ω [rad/s] 5 5 2 x 5 Grid s Voltage Grid s Current 8 x 7 Grid s Power 6.5 8 U net [V] I net [A] 6 P net [W] 2 8 6.5 2 2 Figure 3: Signals evolution of hte considered variables using the Simulink model provided Model_SysMod26.slx and the identified parameters contained in Parameters_HEPP.m All information without warranty.