Question 7 of 8: System Load Demand

Question 7: System Load Demand (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Four sources, each 25 % of maximum system capacity, on an isolated system with no interconnection and no replenishable hydro. The page-16 load curve, read on the hour as a percentage of maximum system capacity:

The curve therefore has a minimum of 27 % at about 03:00, a broad daytime plateau near 65 %, and a sharp evening peak of 88 % at about 19:00. Its mean is 61.3 %, giving a daily load factor of 0.70.

Find. A dispatch for the four sources, hour by hour, that meets the curve, respects the capacity of each source and the closed pumped-storage energy balance, and can be defended on cost and on plant capability.

Assumptions stated (as Note 6 of the paper invites).

• Merit order by marginal cost: nuclear cheapest to run, then coal, then gas. Pumped storage has no fuel cost of its own — its energy cost is the off-peak generation it consumed divided by the round-trip efficiency.
• Pumped-storage round-trip efficiency 75 %, and the machines can pump or generate at the full 25 % rating.
• The nuclear unit is base load and is not manoeuvred: xenon transients and thermal-fatigue limits make daily load-following on a nuclear unit both slow and expensive.
• The coal unit can modulate between about 40 % and 100 % of its rating over minutes to hours, but every cycle costs fuel through a poorer heat rate and costs life through thermal fatigue.
• The gas plant starts and ramps quickly and is the natural load-follower and peaker.
• Spinning reserve, maintenance outages and forced outages are neglected — on a real isolated system the largest unit's capacity would have to be held in reserve, which would change the numbers but not the shape of the answer.

Approach. Fill the load curve from the bottom up in merit order, then size the pumping block so that the energy the reservoir returns over the evening peak is exactly the energy it absorbed overnight, multiplied by the round-trip efficiency.

1. Lay nuclear flat across the whole day. The minimum demand is 27 %, which is above the nuclear unit's 25 % rating, so the unit never has to be backed off. It runs at full output for all 24 hours, contributing $25 \times 24 = 600$ capacity-hours — 41 % of the day's 1470 capacity-hours of demand — at a 100 % load factor. This is the correct home for the plant with the highest capital cost and the lowest fuel cost.
2. Size the pumped storage from the peak it has to cover, not from the water available. With nuclear, coal and gas all at full output the system can supply 75 %, so hydro is needed only where demand exceeds 75 %: from 18:00 to 22:00, requiring 7, 13, 12, 8 and 2 % in successive hours, or 42 capacity-hours in total. Because the reservoir cannot be replenished, that energy must first have been pumped: $$E_{pump} = \frac{E_{gen}}{\eta_{rt}} = \frac{42}{0.75} = \boxed{56\ \text{capacity-hours}}$$
3. Place the pumping load in the deepest part of the trough. Fifty-six capacity-hours over four hours is a pumping load of 14 % of system capacity, and 02:00 to 06:00 is where the curve is lowest. Adding it to the demand gives 49, 41, 41 and 45 % in those hours — all at or below the 50 % that nuclear and coal supply together, so the gas plant stays shut down all night and the pumping is done entirely with nuclear and coal energy. That is the whole point of the exercise: the store converts cheap night-time base-load energy into expensive evening peak energy.
4. Run coal as high as the load allows. Coal now carries the demand between 25 % and 50 % of capacity, plus the pumping block. It sits at its full 25 % for eighteen hours of the day and dips only in the small hours — to a minimum of 15 % of system capacity, that is 60 % of its own rating, comfortably above minimum stable load. Its daily energy is 564 capacity-hours at a load factor of 0.94. Without the pumping block the same unit would have been driven down to 2 % of system capacity at 03:00, far below any coal-fired minimum, and would have had to be shut down and restarted.
5. Give gas the residual. The gas plant fills whatever is left between 50 % and 75 %: it starts at about 07:00, follows the daytime plateau at 14–17 %, ramps to its full 25 % through the evening peak from 17:00 to 23:00, and shuts down overnight apart from a brief 10 % just after midnight. It supplies 320 capacity-hours at a load factor of 0.53 — the lowest utilisation of the four, which is exactly right for the plant with the lowest capital cost and the highest fuel cost.
6. Confirm that the day balances. Generation less pumping must equal demand: $600 + 564 + 320 + 42 - 56 = 1470$ capacity-hours, which is the area under the load curve. The peak hour also fits: at 19:00 demand is 88 % and the dispatch is $25 + 25 + 25 + 13 = 88$ %, so 12 % of installed capacity remains unused as margin.

Reasoning behind the schedule. The order is set by the shape of each plant's cost and by what it is physically able to do. Nuclear has almost all of its cost in capital and almost none in fuel, so every hour it does not run is money already spent and wasted; it is also the plant least able to change output quickly, so it goes at the bottom and stays there. Coal sits next: its fuel is cheap enough to run most of the day but it is the plant that suffers most from cycling, so the pumping block is placed deliberately to keep it loaded overnight rather than to squeeze the last percentage point out of the merit order. Gas is the opposite of nuclear — cheap to build, expensive to run, quick to start — so it takes the variable duty and accepts a poor load factor. The pumped storage is not a generator at all but a broker: it moves 42 capacity-hours from the small hours to the evening, paying 14 capacity-hours of loss for the privilege, and it is worth doing because that same energy would otherwise have had to come from gas at several times the fuel cost, and because without it the coal unit would have to be cycled off every night.

The consequence of the isolation. Because there is no interconnection, the system carries its own reserve and its own regulation: the 12 % of capacity unused at the peak is the margin against a unit trip, and in practice the gas plant would be held part-loaded rather than at its ceiling so that it can pick up frequency. And because the reservoir is off-river, the storage account must balance every day — there is no inflow to draw on if the peak runs long, so the pumping has to be scheduled against the forecast peak with a margin, and a cold snap that stretches the evening peak beyond the stored 42 capacity-hours must be met by gas, not by the hydro.