A. Hammer, H. Götsch, K. Käfer

Transcription

1 CREATING A HYDRAULIC SIMULATION PROGRAMM FOR HYDRO POWER PLANTS IN MATLAB/SIMULINK WITH SPECIAL INTEREST ON THE SURGE TANK BEHAVIOUR AND THE HYDRAULIC INSTABILITY OF PUMP TURBINES A. Hammer, H. Götsch, K. Käfer Abstract: The goal was to create an improved and useroptimized water hammer calculation program, based on an existing program from the Tiroler Wasserkraft AG. Main parts of the improvement were the introduction of the method of characteristics to enhance the accuracy in calculating the water hammer propagation in the pipes and the separated water levels in the surge tank to enable faster deepening for the water level in the shaft.. Now it s possible to simulate pump turbines with a so called S -shape characteristics. By simulating various plants the influence of the rigid surge tank on the simulation showed unexpected water hammer behaviour. 1 Program structure, data Input and stationary start conditions For the simulation are three programs needed. All parameters of the plant and its regulation have to be listed in an Excel form. These values are used in Matlab to iterate the initial stationary condition for the simulation. After this the Simulink model has to be built (Figure 1), by using the corresponding blocks/modules for each section (surge tank, penstock, ) [1]. Figure 1: Simulink model with basic components

2 Iteration of starting conditions To maintain a stable simulation exact starting conditions have to be determined. Therefore a Matlab function called Initialisieren.m iterates the discharge until the hydraulic system is in a stationary state. The iteration stops by the following criteria Q i Q i 1 0,0001 m³/s or i > If there are more than 1000 steps needed the program calls back that the input parameters have to be checked. This function is also used to transfer all given parameters from Excel to the Matlab workspace. Used functions and blocks in Simulink The above mentioned blocks (modules) are masked subsystems. This means that all users are able to tune parameters in the mask of the subsystem but only experienced ones can look behind to adjust the source code or find special signals for the output. So the program remains easy to use and senior users are not restricted in any way. It is possible for every user to look into the embedded Matlab function code where all the calculations take place. This is very useful by testing new functions and debugging. a) b) Figure 2: Blocks in Simulink a) Mask b) Subsystem-penstock Data Output Depending on the aim of the simulation it s possible to add every desired signal to the output. The output uses a To File block which saves all signals routed to it and gives it back as a Time Series Matrix with time vector. Further evaluations have to be done with Matlab. The most likely way is to export all data to Excel make the illustration of diagrams easier. Additionally it s important to get the maximum values (e.g. pressure) all over the pipe length not just chronological sequences. For this reason we made a Matlab function called max_data.m to extract the saved maximum values and connects them to the assigned pipe length. Further post simulation possibilities are to show the pressure oscillation in an animation. For further details please contact the author.

3 2 Simulation of water hammer propagation in penstocks To enhance the accuracy of calculating the hydraulic transients compared to the existing Simulink program of TIWAG we chose the method of characteristics [1]. By taking the elasticity of the pipe, penstock or tunnel into account we derive the wave velocity. The wave velocity and the length of the penstock sections give you the maximum time step for the simulation. Before every simulation the program builds a characteristic grid (Figure 3) for every penstock module to get its calculation points. The grid is equally spaced in the x- and t-plane (Δx, Δt), the correlation is the wave velocity a. x = a t (2.1) Figure 3: Characteristic grid [1] The mathematical aspect behind the method of characteristics is a simplification of a partial differential equation (PDE) to an ordinary differential equation (ODE). This is necessary because the conservation of mass and momentum are PDEs. The linear Combination with the factor C = ±1/a gives you two ODEs, one for the discharge Q and one for the energy head H. These equations are only valid on the skew connection lines of the characteristic grid. Hence all parameters at time t+δt in the penstock can be derived from the values at time t (e.g. Figure 4). the points on the edge are influenced by the boundary conditions, those are the input parameters of the subsystem. Figure 4: Calculation of parameters at the time t+δt

4 3 Separated water levels in the surge tank To simulate the water level in the surge tank in an appropriate way the water levels in the chamber and in the rising shaft must be allowed to separate (Figure 5). Figure 5: Model of the surge tank chamber The discharge between the two systems can be calculated with the formula used for annular spillways (Figure 6). Q is the discharge, µ ü the flow-coefficient and h ü the relative water level above the spillway s edge. This is only valid if the water level in the rising shaft does not influence the flow, otherwise a corrective factor C ü has to be used, derived from the calculation of a flooded weir body (Figure 7) [2]. 2 3 Q = µ 2 2 Ü CÜ U g hü (3.1) 3 Figure 6: Annular spillway a) cup-shaped b) sharp c) flow-coefficient [3] Figure 7: Flooded weir body a) Sketch b) coefficient for imperfect weir [3]

5 Figure 8: Comparison of different programs for surge tank evaluation In Figure 8 you can see the effect of the separated water columns on the results. Compared to the simulations with RSIHS the water level deepens very fast in case of large outflows from the surge tank. The dotted line shows the result of a program for simulating surge tank behaviour. This program can t calculate any transients in the penstock but it is a good benchmark for the used algorithm in the Simulink program.

6 4 The S-shape in pump-turbine characteristics Basic calculation process for the pump-turbine The pump-turbine s behaviour can be taken from the characteristics for flow and torque, e.g. with non-dimensional values Q 11, M 11. These characteristics come from the model test at the manufacturers test rig. By using the characteristics the discharge and torque becomes a function of pressure drop, speed and vane opening (Eq. (4.1)). This works as long there is just one defined value of unit discharge and torque for every value of the unit speed n 11. But if the turbine reaches the region of the S-shape characteristic it finds three possible values (Figure 9). Therefore has to be a routine to manage this problem [2]. Q 11 = Q D 2 H Q = Q 11 (n 11, Y)D² H (4.1) Figure 9: Typical characteristic field for the Unit discharge of a pump turbine Handling the S-shape One way to handle the S-shape is to switch the look up direction from n 11 Q 11 to Q 11 n 11 [ 4]. To switch the way of calculation left me one more problem: the jump over the S-shape in one time step. Mathematically it is possible because in the calculation forward differences are used. This is an explicit way to solve differential equations numerically. The forward calculation leads to inadmissible steps over the S- shape. For the investigated turbine it would have meant a discharge change of about 50 m³/s in a tenth of a second (equals one time step). To keep an eye on that, the characteristics were separated at their reversal points into three sections with a constraint about how to step over to the next section ( Figure 10).

7 Figure 10: Separation in the S-shape With the above shown segmentation of the characteristic it s quite easy to limit the behaviour around the reversal points with some logical requests. E.g. operating point lies in area 1 and tends to jump over in area 3 within the next time step. The program then just allows jumping into area 2. Therefore it is necessary to extrapolate from area 1 into area 2 to get a proper value for the upcoming time step. This procedure is programmed for both reversal points and in both directions. For the simulation with S- shape it is very important to choose a small time step (~0,05 sec.). In practice it is time consuming to take the S-shape into account but results have shown that the simplified models are not conservative (e.g. Figure 11) Headwater pressure [mmh] Speed [%] Discharge [m³/s] Tailwater pressure [mmh] Figure 11: Runaway simulation with S-shape (dotted) and with straighten characteristics (solid)

8 5 Rigid surge tank and its consequence on the simulation At the moment the program works very fine and its results are comparable to measurements and almost congruent to other programs. So the next step was to take a closer look at the surge tank. Actually the simulation of the flow in the surge tank is calculated with the transient Bernoulli equation, thus the water column within the surge tank is dealt with as a rigid body. The problem is that small flow oscillations from the neighbouring elastic pipe act against the inertia of the whole rigid body and produce relatively large pressure changes which on the other hand propagate back into the pipe. This results in very high pressure oscillations in the tunnel because of the lack of damping form the surge tank. In the upcoming investigations we are going to implement a reduced inertia of the water column in the surge tank and compare the results with measurements. Further on the program will be extended with an elastic water column in a surge tank with non-constant length, solved with the characteristic method as in the penstock. References [1] Chaudry, M. H. (1987), Applied Hydraulic Transients, New York, Van Nostrand Reinhold Company [2] Hammer, A. (2011), Dynamische Berechnung von Wasserkraftwerken, Diplomarbeit, TU Wien, Institut für Energietechnik und Thermodynamik [3] Giesecke, (2009), Wasserkraftanlagen Planung Bau und Betrieb, Berlin Heidelberg: Springer Verlag [4] Harbort, T. (1999) Entwicklung eines echtzeitfähigen Simulationsprogramms zur Untersuchung instationärer Vorgänge in Wasserkraftwerken, Dissertation Universität Stuttgart, Institut für Strömungsmechanik und Hydraulische Strömungsmaschinen Authors Andreas HAMMER, TIWAG-Tiroler Wasserkraft AG, 6010 Innsbruck T: ; F: ; Mail: andreas.hammer@tiwag.at Hugo GÖTSCH, TIWAG-Tiroler Wasserkraft AG, 6010 Innsbruck T: ; F: ; Mail: hugo.goetsch@tiwag.at Dr. Klaus Käfer, TU Wien, Institut für Energietechnik u. Thermodynamik, 1060 Wien T: ; F: ; Mail: klaus.kaefer@tuwien.ac.at