Hydraulic turbine characteristics and water starting time

A hydraulic turbine responds to gate movement in an unusual way because of the inertia of the water column in the penstock.

Special characteristics

1. Non-minimum phase behaviour: the turbine transfer function

$$\frac{\Delta P_m(s)}{\Delta G(s)} = \frac{1 - T_w s}{1 + 0.5T_w s}$$

has a zero in the right half of the s-plane ($s = +1/T_w$). So the power first changes in the opposite direction to the gate movement. 2. Initial reverse power: when the gate is suddenly opened, the flow cannot rise at once because of water inertia, so the head (pressure) at the turbine falls and power first drops to −2 times the gate change. It then rises to +1 times the gate change with time constant $T_w/2$: $\Delta P_m(t) = [1 - 3e^{-2t/T_w}]\Delta G$. Closing the gate first gives a power (and pressure) rise. 3. Water inertia governs the response: the response speed is set by the water starting time $T_w$, which is not constant; it rises with load because $U_0$ increases. 4. Need for transient droop: a governor with only permanent droop would be unstable. Hydro governors use a large temporary (transient) droop with a reset time, so they respond slowly at first and are slower than steam unit governors. 5. Water hammer and gate rate limits: gate movements must be slow enough to keep pressure rise in the penstock safe.

 dPm
  1 |                ____________
    |            _.-'
  0 |---.     _-'
    |   |   .'
 -2 |   |_/        (unit step in gate)
    +---+----------------------> t

Water starting time

Water starting time $T_w$ is the time needed for the head $H_0$ to accelerate the water in the penstock from standstill to the velocity $U_0$:

$$T_w = \frac{L\,U_0}{g\,H_0}$$

where $L$ = length of penstock (m), $U_0$ = water velocity (m/s), $H_0$ = head at the gate (m), $g = 9.81$ m/s². It varies with load and is typically 0.5–4 s at full load. A large $T_w$ means a stronger reverse-power effect and harder speed control. A surge tank close to the power house reduces $L$ and so reduces $T_w$.