Open-loop Speed-torque Characteristic

2.5.1 Open-loop Speed-torque Characteristic

The ungoverned or open-loop speed-torque characteristic of a gas turbine has a very steep negative slope and is unsuitable for regulating the power output of the generator. The open-loop characteristic is explicitly determined by the thermodynamic design of the gas turbine, together with the mechani- cal inertial and frictional characteristics of the rotating masses. Without closed-loop feedback control action the initial decline in speed in response to an increase in shaft torque would be mainly deter- mined by the shaft inertia. Let T , ω and P be the torque, speed and shaft power respectively in per-unit terms. The expression relating these variables is,

( 2.45) The open-loop speed-torque function may be expressed as,

P=Tω

ω = f (T )

40 HANDBOOK OF ELECTRICAL ENGINEERING which may be represented by a simple linear function,

( 2.47) where k is a positive number in the order of 1.0 pu equal to the open-loop slope, and ω o is the shaft

ω=ω o − kT

speed at no-load. Reference 7 discusses the slope k in Chapter 2, Section 2.3.1. Assume that the turbine is designed to deliver unit torque at unit speed, therefore,

( 2.48) From which ω o = 1 + k and so (2.47) becomes,

1.0 = ω o − k(1.0) = ω o −k

The speed can now be related to the shaft power rather than the torque,

P=

Or in the form of a quadratic equation,

( 2.51) The two roots of which are,

The positive root applies to the stable operating region, whilst the negative root applies to the unstable region after stalling occurs.

For example assume k = 1.5. Table 2.4 shows the values of the two roots for an increase in shaft power.

Table 2.4. Open-loop steady state speed-power char- acteristic of a gas turbine (k = 1.5)

Shaft power

Shaft speed ω (per unit)

P (per unit)

Positive root

Negative root

1.04 + (unstable)

GAS TURBINE DRIVEN GENERATORS

Table 2.5. Open-loop steady state speed-power characteristic of a gas turbine (k = 0.1)

Shaft power

Shaft speed ω (per unit)

P (per unit)

Positive root

Negative root

At P = 1.0 the torque corresponding to the positive root is T = 0.667 pu, whilst that for the negative root is T = 1.00 pu. Hence the torque at full-load power is less than unity (due to the speed being higher than unity). The above example illustrates the impractical nature of the open-loop speed-torque and speed-power characteristics.

Suppose the design of the engine could be substantially improved such that k could be reduced to say 0.1 (approaching a value for a typical closed-loop feedback controlled system). Table 2.5 shows comparable results to those given in Table 2.4.

It can be seen that unit power is obtained at unit speed in the stable region, and that the stalling point is at a power much greater than unity. The above illustrates more desirable open-loop speed- torque and speed-power characteristics. Unfortunately reducing k to values between say 0.01 and

0.1 by thermodynamic design is not practical. Consequently a closed-loop feedback control system is necessary. Figure 2.12 shows the open-loop speed-power responses for different values of k. The transient response of the gas turbine just after a disturbance in the shaft power is of interest when underfrequency protective relays are to be used to protect the power system from overloading, see sub-section 12.2.10.