Numerical Representation of Hydroelectric Turbines

This section describes the general equations for the schematic turbine shown in Figure 14-13: Schematic of Turbine Hydraulic Element in Hammer (that also shows the upstream and downstream computational points).

Figure 14-13: Schematic of Turbine Hydraulic Element in Hammer

Turbine equations:

H₁+a₁Q=h₁+a₁q₁

H₂+a₂Q=h₂+a₂q₂

Where:	H	=	head at the end of current time step
	Q	=	flow at the end of the current time step
	h	=	head computed during previous time step
	q	=	flow computed during previous time step
		(where a is the wave speed and S is the pipe cross-sectional area)

Pipe head loss equations:

H₁ + f₁Q|Q| = H_C

H₂ + f₂Q|Q| = H_B

Where:	f	=	frictional coefficient
	H_B	=	head at point B at the end of current time step
	H_C	=	head at point C at the end of current time step

Valve head loss equation:

H_C - H_A = K_vQ|Q|

Where:	H_A	=	head at point A at the end of current time step
	K_V	=	valve loss coefficient

Four-quadrant turbine curves:

M_hyd = F_M(Q,N,w)

H_A - H_B = F_H(Q, N, w)

Where:	M_hyd	=	hydraulic torque
	N	=	rotational speed of the turbine
	w	=	wicket gate function
	F_M	=	torque function
	F_H		head function

Conservation of angular momentum:

Where:	n	=	turbine's rotational speed computed during previous time step
	m	=	torque computed during previous time step
	M	=	torque at current time step
		(W= weight of turbine and generator, R= radius fo gyration)

Algebraic manipulations reduce equations () to () to a pair of non-linear equations in the unknowns Q and N as follows:

F_H(Q, N, w) - (K_v + f₁ + f₂)Q|Q| - (a₁ + a₂)Q - (h₂ - h₁ + a₂q₂ + a₁q₁) =0

The non-linear equations () and () can be solved by iteration using Newton's method in conjunction with the four-quadrant head and torque curves for various wicket gate positions.