Power Plant

Pump-turbine

The operation of a pump-turbine is characterized by the operating power. By convention, the power is positive if the machine is in turbine mode (supplied power) and negative if it is in pump mode (used power). In turbine mode, a machine can operate in a power range between $P_{min}^T$ and $P_{max}^T$ ($0< P_{min}^T \leq P_{max}^T$). Similarly, in pump mode, the power range lies between $P_{max}^P$ and $P_{min}^P$ ($P_{max}^P \leq P_{min}^T < 0$).  Therefore, a machine can be in 3 possible state : either it is in pump mode, or in turbine mode or switched off. Its operating power is hence described by:

$P \in [P_{max}^P, P_{min}^P] \cup \{0\} \cup [P_{min}^T, P_{max}^T]$

The operating power of a pump turbine can not be changed within one hour so as not to excessively damage the blades of the machine.

Tanks

In the remaining, we suppose that the two tanks are cuboids of identical dimensions $L \times l \times H$ without leaks. Notice that all the pump-turbines are connected to the two same tanks. In order to avoid floods, the upper and the lower tanks should never overflow. Under this constraint, one can notice that the repartition of the water in the two tanks is completely described by the height of the waterfall:

$h=h_{sup} − h_{inf} + \Delta H$

The flow of water set in motion by the pump-turbine is related to the power with the following so-called operating equations:

• $P = \alpha_P Q$ in pump mode
• $P = \alpha_T Q$ in turbine mode
• $P =Q=0$ when switched off

where $P$ is the power in MW and $Q$ the water flow in $m^3.s^{−1}$. The constants $\alpha_T, \alpha_P \in \mathbb{R}^+$ are related to the machine efficiency (yield) and to physical constants. The sign convention on the power implies that the water flow is positive when the water goes down and it is negative when the water goes up.

Questions to help you:
(a) Calculate the variation of fall height between time $t$ and $t + 1$ knowing that the pump-turbine is operating at power $P$ .
(b) Write this equation with the variables previously defined in the model.

Cost of operating mode changes

Changing the operating mode of a pump-turbine needs a manipulation which has a cost depending of the initial and of the final operating mode. We note $c_{AT}$ and $c_{AP}$ the cost induced by passing from the switched off mode $A$ to the turbine mode $T$ and from the switched off mode $A$ to the pump mode $P$ respectively. Similarly, we define $c_{TA}$ and $c_{PA}$ the costs induced by the reverse processes. A machine can change directly from the pump mode to the turbine mode but it must necessarily be switched off when passing from a mode to the other. Although those changes are complex we suppose that they are instantaneous.

Cooling of the pump-turbines

Like any electrical device, a pump heats up when it is running. To avoid overheating and damaging the components, it is necessary that it never runs $R=12$ hours in a row. Thus on any interval of 12 hours, a pump-turbine must spend at least one hour idle.