Q1Power System 2

Question

Q.1. Two generators rated 200 MW and 400 MW are operating in parallel. The droop characteristics of their governors are 4% and 5%, respectively from no load to full load. The speed changers are so set that the generators operate at 50 Hz sharing the full load of 600 MW in the ratio of their ratings. If the load reduces to 400 MW, how will it be shared among the generators and what will the system frequency be? Assume free governor operation.

Answer 1

Given Data

Generator A: rated 200 MW, governor droop RA = 4%
Generator B: rated 400 MW, governor droop RB = 5%
Initially operating at 50 Hz, sharing 600 MW total load in the ratio of their ratings (i.e., PA=200 MW, PB=400 MW, both fully loaded at 50 Hz)
Load reduces to 400 MW total; find new sharing and system frequency, assuming free governor action (no speed-changer adjustment)

Step 1: Determine No-Load Frequency Set for Each Governor

A governor's droop characteristic defines how much system frequency must fall, from its no-load value f0, to reach full-load output at the machine's rated MW, over the machine's full 0-to-rated-power range. A droop of R percent corresponds to a full-range frequency drop of R percent of the nominal (50 Hz) frequency:

Δ f_{A} = R_{A} \times f_{n o mina l} = 0.04 \times 50 = 2 Hz (over 0 to 200 MW range)

Δ f_{B} = R_{B} \times f_{n o mina l} = 0.05 \times 50 = 2.5 Hz (over 0 to 400 MW range)

Since both generators are operating at their full rated output (PA=200 MW=rating, PB=400 MW=rating) at the given operating frequency of 50 Hz, this corresponds to each machine's full-load point on its own droop line, so the no-load frequency (the frequency intercept at zero output, as set by each machine's speed changer) for each is:

f_{0, A} = 50 + Δ f_{A} = 50 + 2 = 52 Hz

f_{0, B} = 50 + Δ f_{B} = 50 + 2.5 = 52.5 Hz

Step 2: Set Up Droop Equations for the New Operating Point

With free governor operation (speed-changer settings, and hence f0,A and f0,B, unchanged) and the load reduced to 400 MW total, both generators must settle to a new common system frequency f, with each generator's output related to f by its own linear droop line:

P_{A} = \frac{f _{0, A} - f}{Δ f _{A}} \times 200, P_{B} = \frac{f _{0, B} - f}{Δ f _{B}} \times 400

with the constraint that the total generation must equal the new total load:

P_{A} + P_{B} = 400 MW

Step 3: Solve for Common Frequency f

Substituting the droop expressions into the power balance equation:

\frac{( 52 - f )}{2} \times 200 + \frac{( 52.5 - f )}{2.5} \times 400 = 400

100 (52 - f) + 160 (52.5 - f) = 400

5200 - 100 f + 8400 - 160 f = 400

13600 - 260 f = 400 \Rightarrow f = \frac{13200}{260} = 50.77 Hz

Step 4: Determine New Load Sharing

P_{A} = \frac{52 - 50.77}{2} \times 200 = \frac{1.23}{2} \times 200 \approx 123.08 MW

P_{B} = \frac{52.5 - 50.77}{2.5} \times 400 = \frac{1.73}{2.5} \times 400 \approx 276.92 MW

Check: PA+PB = 123.08+276.92 = 400.00 MW, matching the reduced total load exactly, confirming the solution's consistency.

Result: the new system frequency settles at approximately 50.77 Hz, with Generator A (200 MW rating) supplying approximately 123.08 MW and Generator B (400 MW rating) supplying approximately 276.92 MW. Notably, the new sharing ratio (123.08:276.92, approximately 1:2.25) is close to, but no longer exactly equal to, the original 1:2 rating ratio, since Generator B, having the larger droop percentage (softer/more sloped droop characteristic, in absolute Hz/MW terms scaled by its larger rating), reduces its output somewhat less proportionally than Generator A does as system frequency rises above 50 Hz — this illustrates how, in free-governor operation without any manual speed-changer readjustment, actual load sharing among parallel generators is governed jointly by each unit's rating and its specific per-unit droop setting, not simply by the units' rated capacity ratio alone.

It is worth noting that this new sharing arrangement remains entirely consistent with each generator's own fixed droop characteristic, meaning no manual intervention or speed-changer adjustment was required for the system to settle at this new operating point purely through the natural, automatic response of each machine's governor to the common system frequency change. In a practical power system, if the operator instead wished to restore the original 1:2 (rating-proportional) sharing ratio even after the load change, or to bring system frequency back exactly to 50 Hz, a secondary control action (automatic generation control, adjusting the speed-changer setpoints of one or both units) would be required, since free-governor action alone can only establish a new equilibrium consistent with the machines' existing droop settings and cannot, by itself, simultaneously restore both the nominal frequency and the originally-desired proportional sharing ratio following a load change of this kind.