A salient-pole synchronous generator with saturated synchronous reactances Xd-0.78 per unit and Xq-0.63 per unit is connected to a rated-voltage infinite bus through an external impedance Xbus- 0.09 per unit. a. Assuming the generator to be supplying only reactive power (i) Find minimum and maximum per-unit field excitations (where 1.0 per unit is the field current required to achieve rated open-circuit voltage) such that the generator does not exceed its rated terminal current. (ii) Using MATLAB, plot the armature current as a function of field excitation as the field excitation is varied between the limits determined in part (i). b. Now assuming the generator to be supplying 0.25 per unit rated real power, on the same axes add a plot of the per-unit armature current as a function of field excitation as the field current is varied in the range for which the per-unit armature current is less than 1.0 per unit. c. Repeat part (b) for generator output powers of 0.50 and 0.75 per unit. The final result will be a plot of V-curves for this generator in this configuration.

ANSWER:

a. Minimum and maximum per-unit field excitations

To determine the minimum and maximum per-unit field excitations, we need to find the values of field excitation that result in rated terminal current. The rated terminal current can be calculated using the following equation:

It = V/√(Xd² + Xq²)

where:

  • It is the rated terminal current (per unit)
  • V is the rated voltage (per unit)
  • Xd is the direct-axis synchronous reactance (per unit)
  • Xq is the quadrature-axis synchronous reactance (per unit)

In this case, the rated terminal current is:

It = 1.0 / √((0.78)² + (0.63)²) = 0.97 per unit

The minimum per-unit field excitation is the value that results in a terminal current of 0.97 per unit when the generator is operating at zero real power. The maximum per-unit field excitation is the value that results in a terminal current of 0.97 per unit when the generator is operating at its maximum real power.

To find the minimum per-unit field excitation, we can use the following equation:

Et = V / √((Xd² + Xq²) + (Xbus/2)²)

where:

  • Et is the per-unit field excitation
  • Xbus is the external impedance (per unit)

In this case, the minimum per-unit field excitation is:

Et = 1.0 / √((0.78)² + (0.63)² + (0.0)

MATLAB SOLUTION TO SOLVE PROB

% Define parameters
Xd = 0.78;
Xq = 0.63;
Xbus = 0.09;
ratedVoltage = 1.0;
ratedCurrent = ratedVoltage / Xd;

% Part (a)

% Minimum field excitation
minimumFieldEx = 0.32;
Vs = ratedVoltage * minimumFieldEx;
Is = Vs / (Xd + Xbus);

while Is > ratedCurrent
    minimumFieldEx = minimumFieldEx + 0.01;
    Vs = ratedVoltage * minimumFieldEx;
    Is = Vs / (Xd + Xbus);
end

% Maximum field excitation
maximumFieldEx = 1.5;
Vs = ratedVoltage * maximumFieldEx;
Is = Vs / (Xd + Xbus);

while Is > ratedCurrent
    maximumFieldEx = maximumFieldEx - 0.01;
    Vs = ratedVoltage * maximumFieldEx;
    Is = Vs / (Xd + Xbus);
end

% Field excitation range
fieldExRange = linspace(minimumFieldEx, maximumFieldEx, 100);

% Armature current as a function of field excitation
Is = ratedVoltage * fieldExRange / (Xd + Xbus);

% Plot armature current
plot(fieldExRange, Is);
xlabel('Field excitation (per unit)');
ylabel('Armature current (per unit)');
title('Armature current as a function of field excitation');

% Part (b)

% Generator output power of 0.25 per unit
P = 0.25;
Q = P * Xd / (Xd^2 - Xq^2);
V = ratedVoltage;
Vs = V + j * Q * Xd;
Is = Vs / (Xd + Xbus);

% Field excitation range for armature current less than 1.0 per unit
fieldExRange = linspace(Vs / ratedVoltage, maximumFieldEx, 100);
Is = Vs / (Xd + Xbus);

while Is > 1.0
    fieldExRange = fieldExRange(2:end);
    Is = Vs / (Xd + Xbus);
end

% Armature current as a function of field excitation
Is = Vs / (Xd + Xbus);

% Plot armature current
hold on;
plot(fieldExRange, Is);

% Part (c)

% Generator output power of 0.50 per unit
P = 0.50;
Q = P * Xd / (Xd^2 - Xq^2);
V = ratedVoltage;
Vs = V + j * Q * Xd;
Is = Vs / (Xd + Xbus);

% Field excitation range for armature current less than 1.0 per unit
fieldExRange = linspace(Vs / ratedVoltage, maximumFieldEx, 100);
Is = Vs / (Xd + Xbus);

while Is > 1.0
    fieldExRange = fieldExRange(2:end);
    Is = Vs / (Xd + Xbus);
end

% Armature current as a function of field excitation
Is = Vs / (Xd + Xbus);

% Plot armature current
plot(fieldExRange, Is);

% Generator output power of 0.75 per unit
P = 0.75;
Q = P * Xd / (Xd^2 - Xq^2);
V = ratedVoltage;
Vs = V + j * Q * Xd;
Is = Vs / (Xd + Xbus);

% Field excitation range for armature current less than 1.0 per unit
fieldExRange = linspace(Vs / ratedVoltage, maximumFieldEx, 100);
Is = Vs / (Xd + Xbus);

while Is > 1.0
    fieldExRange = fieldExRange(2:end);
    Is = Vs / (Xd + Xbus);
end

% Armature current as a function of field excitation
Is = Vs / (Xd + Xbus);

% Plot armature current
plot(fieldExRange, Is);

% Label axes and title
xlabel('Field excitation (per unit)');
ylabel('Armature current (per unit)');
legend('0.25 per unit output power', '0.50 per unit output power', '0.75 per unit output power');
title('V-curves for the synchronous generator');

Leave a Comment