Water turbine transfer function and characteristics

The water turbine transfer function relates a small change in gate position to the change in mechanical power. Because the water column in the penstock has inertia, the turbine is a non-minimum phase system.

Derivation

Assumptions: lossless turbine, incompressible water, rigid penstock, and small changes about an operating point.

Velocity of water in the penstock: $U = K_u\,G\sqrt{H}$, where $G$ = gate opening and $H$ = head at the gate.

Turbine power: $P_m = K_p\,H\,U$.

Linearising and normalising by initial values ($U_0, G_0, H_0, P_{m0}$), with bar for per-unit change:

$$\begin{aligned} \Delta\bar U &= \tfrac{1}{2}\Delta\bar H + \Delta\bar G \\ \Delta\bar P_m &= \Delta\bar H + \Delta\bar U \end{aligned}$$

Eliminating $\Delta\bar H = 2(\Delta\bar U - \Delta\bar G)$:

$$\Delta\bar P_m = 3\Delta\bar U - 2\Delta\bar G$$

Acceleration of the water column (length $L$, area $A$): the net force from the head change accelerates the water mass,

$$\rho L A\,\frac{d\Delta U}{dt} = -A\rho g\,\Delta H \;\Rightarrow\; T_w\,\frac{d\Delta\bar U}{dt} = -\Delta\bar H$$

where the water starting time is

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

In Laplace form: $T_w s\,\Delta\bar U = -2(\Delta\bar U - \Delta\bar G)$, so

$$\frac{\Delta\bar U}{\Delta\bar G} = \frac{1}{1 + \tfrac{1}{2}T_w s}$$

Substituting in $\Delta\bar P_m = 3\Delta\bar U - 2\Delta\bar G$:

$$\frac{\Delta\bar P_m(s)}{\Delta\bar G(s)} = \frac{3}{1 + 0.5T_w s} - 2 = \frac{1 - T_w s}{1 + 0.5T_w s}$$

Special characteristics of a hydraulic turbine

1. Non-minimum phase response: the transfer function has a zero in the right-half plane at $s = 1/T_w$. Power initially changes opposite to the gate movement. For a unit step in gate opening, power first falls to $-2$ pu and then rises to $+1$ pu with time constant $T_w/2$:

$$\Delta\bar P_m(t) = 1 - 3e^{-2t/T_w}$$

2. Water inertia decides the response. $T_w$ is typically 0.5–4 s and changes with load, because $U_0$ and $H_0$ change.
3. Governor needs transient droop: a fast governor with normal droop would be unstable. So hydro governors have a large temporary droop with a reset time (or slow integral action), making hydro units slower to respond than steam units.
4. Water hammer: quick gate closing causes pressure rise; gate speed must be limited.
5. Hydro turbines can start and pick up load quickly from standstill, so they are good for peaking and frequency control once the above is allowed for.