Project Grade 11 Math
Varnwell's Pumping Schedule: Least Cost Under a Time-of-Use Tariff
A four-session grade 11 project where teams build an hourly tank balance, design a 24-hour pumping schedule under storage limits, and defend its cost to a board panel.
Objectives
- Students will build an hourly running balance of a storage tank from demand and pumping.
- Students will express storage and capacity limits as inequalities and test schedules against them.
- Students will compare the energy cost and the demand charge of competing schedules.
Vocabulary
- Pump station
- A pump station is a building full of powerful pumps that push water through pipes or lift it uphill. Pumps give water the extra push that gravity alone cannot provide.
Also: time-of-use tariff, demand charge, usable storage, running balance
Teacher overview
Electricity is one of the largest line items a water utility pays, and utilities that buy power on a time-of-use tariff can cut that bill substantially by moving their pumping into the cheap hours and letting storage carry the expensive ones. The catch is that storage is finite, so a schedule that avoids every costly hour will empty the tank before the evening peak ends. Over four sessions, teams design a 24-hour pumping schedule for the fictional Varnwell Water District that meets every hour's demand, keeps the Ulverby tank inside its limits, and costs as little as possible. The part teams most often mishandle is the demand charge, which is set once a month by the highest simultaneous draw rather than by total energy, so the saving from running both pumps to reach the cheap hours has to be weighed against a month of that charge rather than against a single day's energy bill.
Project Brief
Driving question. How should the fictional Varnwell Water District run its pumps over 24 hours so that customers are served, the tank stays safe, and the monthly power bill is as low as it can be?
Your client. The fictional Varnwell Water District board has asked your team for a written pumping schedule and a short defense of it. Your work goes to a board panel in the final session.
The pump station. The fictional Hackwell Pump Station has two identical pumps. Each delivers 700 gpm, which is 42,000 gallons per hour, and each draws 55 kW while running. In any hour you may run zero, one, or two pumps. A pump runs for a whole hour or not at all.
The tank. The Ulverby tank holds 350,000 gallons when full. It must never fall below 70,000 gallons, which is the reserve the district keeps for fire flow. It cannot hold more than 350,000 gallons. The tank must end the 24 hours holding at least as much as it held at hour 0.
The tariff. Power costs $0.26 per kWh during the on-peak hours, which run from hour 14 through hour 19. All other hours cost $0.08 per kWh. On top of the energy charge, the district pays a demand charge of $13 for each kilowatt of the highest simultaneous draw recorded in the month.
The demand. Hourly demand for the district is fixed and is given below in gallons per hour.
| Hours | Demand each hour (gal) |
|---|---|
| 0 through 5 | 16,000 |
| 6 through 8 | 58,000 |
| 9 through 13 | 40,000 |
| 14 through 17 | 50,000 |
| 18 through 19 | 82,000 |
| 20 through 23 | 36,000 |
What you turn in. A written proposal of two to three pages containing a 24-hour schedule table with one row per hour, a graph of tank level against hour, the daily energy cost split into on-peak and off-peak, the monthly demand charge, one paragraph explaining why no cheaper schedule exists, and one paragraph naming the risk your schedule carries if demand runs higher than the table. Each team also gives a five-minute presentation to the board panel.
Milestones
Session 1, build the balance. Set up a 24-row table with columns for the hour, the demand, the gallons pumped, and the tank level at the end of the hour. Choose a starting level and test two schedules that you already expect to fail: pumping in no hour at all, and pumping both pumps in every hour. Record where each one breaks a limit. Then find the total demand for the day and divide it by 42,000 to find the pump-hours the day requires. Turn in the table and the pump-hour figure.
Session 2, write the constraints and draft a schedule. Write the storage floor, the tank ceiling, and the end-of-day condition as three inequalities on your running balance. Write tank level at the end of hour n as an equation in the hours pumped so far and the demand so far. Draft one schedule that satisfies all three inequalities and graph its tank level against the hour on labeled axes. Turn in the three inequalities, the equation, the schedule, and the graph.
Session 3, cost it and cut it. Compute your schedule's on-peak energy, off-peak energy, and demand charge separately. Then find the smallest number of on-peak pump-hours any legal schedule can use, and justify that number from the on-peak demand and the usable storage. Build a second schedule that reaches that minimum and cost it the same way. Turn in both cost breakdowns and a sentence saying which schedule wins and by how much.
Session 4, present and defend. Give a five-minute presentation to the board panel with your tank-level graph displayed. State your schedule, your monthly cost, and your justification that no cheaper schedule exists. Answer two questions from the panel, one of which will ask what happens on a hot day when every hour's demand runs one-tenth higher. Turn in the final proposal after the presentation.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Rubric
Score each criterion on the four levels below.
| Criterion | 4, exceeds | 3, meets | 2, approaching | 1, beginning |
|---|---|---|---|---|
| Hourly balance | All 24 rows are arithmetically correct, the running level follows from the row above, and the starting level is stated and justified. | The balance is correct with one or two arithmetic slips that do not change the verdict on any limit. | The table is built but the running level is recomputed inconsistently, or several rows do not follow from the row above. | No usable running balance, or demand and pumping are not tracked hour by hour. |
| Constraints as inequalities | The floor, the ceiling, and the end-of-day condition are each written correctly as inequalities and each is tested against the schedule at the hour where it binds. | All three inequalities are written correctly and tested, with a minor gap in the testing. | Some limits are stated in words only, or one inequality is missing or reversed. | Limits are mentioned but never written mathematically or tested. |
| Cost analysis | On-peak energy, off-peak energy, and the demand charge are computed separately and correctly, and the comparison between the two schedules is stated in dollars per month. | All three cost pieces are computed with a small error, and the comparison is stated. | Energy cost is computed but the demand charge is omitted or misapplied, or the two schedules are not compared. | Costs are asserted without a calculation a reader can follow. |
| Argument and defense | The minimum on-peak pumping is justified from the on-peak demand and the usable storage, the graph is labeled and readable, and the hot-day risk is answered with the team's own numbers. | The justification is present and mostly correct, the graph is labeled, and the risk question is answered. | The proposal claims the schedule is cheapest without supporting it, or the graph lacks labels. | The presentation restates the schedule without explaining why it was chosen. |
Differentiation
Support
Give these teams a partly built balance table for the fictional Varnwell day, with the demand column filled and the first six rows completed, so their work begins at the decision of when to pump. Provide the three inequalities already written, and let them evaluate two supplied schedules against the limits before designing one of their own.
Extension
Ask these teams to treat the fictional Ulverby tank size as a variable and find the smallest tank that would allow the district to avoid all on-peak pumping. Then have them compare the yearly saving from that change against a construction cost the team estimates from research and labels as a fictional figure, and report whether the larger tank pays for itself.
ELL supports
Pre-teach "demand," "peak hours," "storage," and "demand charge" with a labeled drawing of a pump station filling a tank while houses draw from it. Provide a schedule table template with the column headings already written, and presentation sentence frames such as "We pump ___ pumps during hours ___ because ___," and "Our monthly cost is ___, which is ___ than the schedule we rejected."
Extensions
Teams can extend the fictional Varnwell schedule across a seven-day week in which weekend demand runs lower, and report whether the cheapest weekday schedule stays cheapest. A second task is to describe how a district would verify its own hourly demand in practice, and why a schedule built on a single day's table carries risk.