Ask the solver, then check its answer
HiGHS in your browser: solve, check, and read prices
- Modeler · L3
- Lesson 11 of 14
- 15 min
- After lesson 10
- Completed
3 September 2025: the solver's plan for Prairie Town's day
With JavaScript on, you run the solver yourself (about 3.5 MB, loaded once), try to beat it and check its answer.
Data: lesson 0's day. Demand, wind and solar follow the real Southwest Power Pool shapes of 3 September 2025 (EIA-930), scaled to the town.
Illustrative: Prairie Town, its plants, costs and the unserved-energy penalty. Solver: HiGHS (MIT licence), on your device.
What you just did, as diagrams
- Solver
- Software that finds the best plan of a model, or proves that none exists; here HiGHS, in well under a second once it has loaded.
- Checks
- Balance, limits, ramps, energy and money: a solver answers the model you wrote, so check it against the world.
- Shadow price
- How much the best cost rises per small extra MWh of demand, away from a kink; $40 in 18 of 24 hours, where the combined-cycle is the marginal plant.
- Mixed-integer programs
- Their solve returns no ordinary shadow prices; prices come from a linear program with the on-or-off choices fixed (lesson M2).
Go deeper · inside the solverPeek inside: the simplex method walks the corners
Lesson 10's hour again. The simplex method starts at a corner and moves along an edge to a neighbouring corner that is never worse, again and again, until no neighbour is cheaper. In a linear program that corner is the best plan overall. On this hour every move is cheaper.
With JavaScript on, step through the walk here.
Go deeper · ModelerWhy a mixed-integer solve returns no shadow prices
- Linear program. As demand changes, the best cost is continuous and piecewise linear: straight pieces joined at kinks. The slope of a piece is the shadow price.
- Mixed-integer program. The solver searches over on-or-off choices (an off peaker, or one at 30 MW or more), so it returns no ordinary LP shadow prices. The best cost can also jump when a plant switches on.
| At a 30 MW/h ramp limit | Linear program | Mixed-integer program |
|---|---|---|
| Best cost for the day | $106,300 | $107,200 |
| Peaker | 35 MWh, some hours below 30 MW | off all day; more wind and solar curtailed |
| Shadow prices from the solve | one per hour | none |
To price a mixed-integer plan, fix its unit commitment (which plants are on), re-solve the linear program, and read prices conditional on that commitment. Lesson M2 builds that model.
Go deeper · ModelerWhen a shadow price is not unique
Away from a kink, the shadow price is the cost change per small extra MWh. At a kink, compare the increase and the decrease separately: any value between them is a valid shadow price.
| Hour | +1 MWh costs | −1 MWh saves | HiGHS reports |
|---|---|---|---|
| 07:00 | $100 | $80 | $80 |
| 09:00 | $40 | $20 | $40 |
- 07:00. Up: the peaker makes the MWh, since the ramp rule would lift 08:00 and 09:00 too. Down: 07:00 and 08:00 each fall by 1 MWh.
- 09:00. Up: the combined-cycle. Down: 07:00, 08:00 and 09:00 each fall by 1 MWh ($120), and the peaker covers 07:00 (−$100).
Such optima are called degenerate. They are common in real models, so read a single reported price with care.
Each price here is a marginal value under stated assumptions: fixed plant sizes, a linear model, and Prairie Town as an island. It is the merit-order price of lesson 8 read from a solver, not a forecast of a market price.
Check yourself
Myth or fact? “If the solver says ‘optimal’, the model is right.”
- Myth
- Fact
Answer
Myth. The sign-error model was ‘optimal’ too, at $52,160. Optimal means best for the model as written; checks tell you whether the model is right.
Myth or fact? “The demand rule's shadow price always equals the running cost of a plant that is running.”
- Myth
- Fact
Answer
Myth. Where a ramp rule is at its limit it is $0 or $80 here: a change at one hour moves the plan in a neighbouring hour too, and the price counts both.
Myth or fact? “In a linear program, a plan that no small change can improve is the best plan overall.”
- Myth
- Fact
Answer
Fact. A linear program has no ‘local’ best that is not also the global best. That is why you could not beat the solver.
Myth or fact? “A mixed-integer program gives shadow prices just like a linear program.”
- Myth
- Fact
Answer
Myth. A mixed-integer solve returns no ordinary shadow prices. Prices need a second step: fix the on-or-off choices, re-solve the linear program, and read prices conditional on them.
At 19:00 the shadow price of the demand rule is $40/MWh. A new 5 MW load runs for that hour. By about how much does the best cost rise?
- $40
- $200
- $500
- $0
Answer
About $200: 5 MWh × $40. A shadow price holds for small changes that leave the same rules binding (exactly at their limits).
A model reports a day's cost far below what you expected. What should you do first?
- Celebrate the saving
- Run a different solver
- Check that supply equals demand every hour with the reported plan
- Lower the time limit
Answer
Check the plan against the physics. Another solver would solve the same wrong model just as well.
Numbers, sources and what’s next
Numbers to remember
| Number | What it means | Source |
|---|---|---|
| $104,000 | The best total cost of lesson 0's day, with nothing unserved: $400 below the cheapest-first rule's $104,400. The model has 96 decisions and 72 rules. | This lesson's model (illustrative) |
| $40/MWh | The demand rule's shadow price in 18 of 24 hours: the combined-cycle's running cost. | This lesson's model (illustrative) |
| MIT | HiGHS, the solver on this page, is open source under the MIT licence; it solves linear programs by simplex, interior point or first-order methods, and mixed-integer programs by branch-and-cut. | HiGHS documentation · checked 2 Nov 2026 |
| 5 minutes | How often the western Energy Imbalance Market dispatched generating units, as EIA reported in July 2017. | EIA, Today in Energy · published 14 Jul 2017 |
Next on the starter path
In the real world
Market operators re-solve dispatch models like this one every few minutes: 5 in the western Energy Imbalance Market, as EIA reported in 2017.
Their prices are shadow prices of balance rules, one for each location on the network: lesson M1.
Sources and further reading
- HiGHS. Documentation: the open-source linear and mixed-integer solver used on this page.
- EIA (2017). The western U.S. sub-hourly wholesale electricity market is expanding. Today in Energy.
- EIA. Hourly Electric Grid Monitor (Form EIA-930): the Southwest Power Pool data behind Prairie Town's day.
- JuMP. Modelling language for mathematical optimization, with HiGHS among its solvers.
- Earlier lessons: lesson 0 (the day by hand), lesson 10 (the model written down), lesson 8 (the merit-order price).
Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.
Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.