Aim of the task

The aim of the task is to learn about limitations related to admissible operating states of a generating source based on a synchronous generator. The area of admissible operating states will be determined by means capability chart.

Introduction

Real and reactive power produced by the generator depends on [1]:

1. the electromagnetic force (emf)    which is proportional to the generator field current  ,
2. voltage U on the terminals of the step-up transformer,
3. power angle  (where – the angle  between the voltage U and the generator emf ).

The voltage of a generator operating in a power system cannot change much and must be held within typically 10% percent of the network rated voltage.

Limits in the generator real and reactive power control are the result of the following constructional and operational constraints:

1. Stator (armature) current I must not cause overheating of the armature winding. Hence it must be smaller than a certain maximum value , i.e. .
2. Rotor (field) current   must not cause overheating of the field winding. Hence it must be smaller than a certain maximum value , i.e.  or  .
3. The power angle must not be higher than a maximum value due to stable generator operation  .
4. The temperature in the end region of the stator magnetic circuit must not exceed a maximum value
5. The generator real power must be within the limits set by the turbine power, i.e. .

Capability chart

Capability chart of the synchronous generator can be determined analysing the following conditions:

a) Condition (1) – stator (armature) current

On the P-Q plane, that condition (i) corresponds to circle with the radius   and the centre at the origin. Assuming a given voltage U and loading  one gets

which corresponds to a circle with a radius  .

b) Condition (2) – rotor (field) current

In the P-Q plane, it corresponds to a circle with with a radius    and a centre displaced from the origin along the reactive power axis Q by the value .

c) Condition (3) – the power angle

On the P-Q plane, condition (iii) concerning the maximum value of the power angle, corresponds to a straight line. The position and tangent of the line can be derived the following way:

Substituting   leads to    where          and     

Above equation describes a straight line intersecting the reactive power axis at the angle   and at the point: or  .

d) Condition (4) – the temperature

There is no simple mathematical formulation describing  the constraint corresponding the end region heating limit. The relevant curve has to be determined experimentally by the manufacturer.

e) Condition (5) – turbine power

For steam turbines, the upper constraint   is due to the maximum (rated) output of the turbine while the lower constraint   is due to stable operation of burners at a low turbine output. On the P-Q plane the upper and lower limits correspond to straight vertical lines   and .

Fig. 6.1 shows the synchronous generator capability chart determined for given voltage.

Fig.6.1. Synchronous generator capability chart assuming a given voltage, where segment A-B  corresponds to      the power angle condition,  segment B-C corresponds to the temperature condition, segments C-D and E-F corresponds to stator (armature) current condition, segment D-E corresponds to upper turbine power condition, segment G-A corresponds to lower turbine power condition [1]

Exercise

Content and input data:

Analysis refers to the synchronous generator described in Task 5.

To do:

Calculate and plot a capability chart of a synchronous  generator.

In the chart, mark the point corresponding to load of the generator. Verify if the operation point of the generator (P, Q) is located in the admissible area.

Additional assumptions:

• Calculation should be carried out in per units (base values correspond to rated values of the generator U_b= U_N =15,75 kV, S_b= S_N =250MVA)
• maximum value of the stator (armature) current is equal to current rated value I_MAX=I_n
• turbine power limits: P_MAX=1,05 P_n, P_min=0,55 P_n or 0,45 P_n (where P_n – generator rated real power)
• generator-rated power factor cosjn = 0,85
• maximum value of excitation E_fMAX = E_qMAX = 3,5 or 3,6 or 3,7 or 3,8 or pu (for base values corresponding to rated values of the generator)
• power angle limit: = 120 or 110 or 100 deg
• Condition corresponding to temperature in the end region of the stator magnetic circuit is ignored in analyze.

Hint:

In Task 5 reactances of synchronous generator were expressed in per unit but the base values were: U_b= 220 kV, S_b=100MVA (network base values). For capability chart calculation the base values related to rated values of generator U_n and S_n should be taken into account.

Recalculation of the generator reactance:

Z_b1 – base impedance for U_b =220 kV and S_b=100 MVA

X_d1 – generator reactance X_d from task 5

Z_b2 – base impedance for U_b =15,750 kV and S_b =250 MVA

 – transformer ratio ( 245/15,75  or = 250/15,75 )

References

[1] Machowski J., Bialek J, Bumby J.,  Power system dynamics: stability and control. New York, USA: John Wiley & Sons; 2020.