Economic Dispatch
Linear / Quadratic Programming · Polynomial
Economic dispatch is solved every 5–15 minutes in real electricity markets, with LP-based solutions saving utilities 1–5% of annual fuel costs (Hobbs et al., 2001). At each dispatch interval, a grid operator must decide how much power each online generator should produce to meet total system demand at minimum fuel cost. This is economic dispatch — a convex QP when fuel cost curves are quadratic, solvable exactly by equal incremental cost (λ-iteration) or LP simplex.
Where This Decision Fits
Energy systems planning chain — the highlighted step is what this page optimizes
The Problem
Minimize generation cost while meeting system demand
A regional grid operator manages 6 generators supplying a total system demand of 2,100 MW. Each generator has a minimum and maximum output level, a fuel type, and a marginal cost per MWh. The operator must allocate generation across all online units to minimize total hourly fuel cost while satisfying the demand balance constraint and respecting each unit’s capacity bounds.
Nuclear and coal plants have minimum generation constraints — they cannot ramp below a threshold due to thermodynamic and safety considerations. Renewable sources (wind, solar) have near-zero marginal cost but variable capacity depending on weather conditions.
subject to
Σi pi = D // demand balance
Pimin ≤ pi ≤ Pimax // generator capacity bounds
pi ≥ 0 // non-negativity
Where ci is the marginal cost of generator i ($/MWh), pi is the power output, D is total demand, and Pimin, Pimax are the operating limits.
See Linear Programming theoryTry It Yourself
Dispatch power across generators to minimize fuel cost
Economic Dispatch Optimizer
6 Generators · 2,100 MW| Generator | Type | Min (MW) | Max (MW) | Cost ($/MWh) | Online |
|---|
Ready. Click “Solve & Compare All Algorithms” to run.
| Algorithm | Total Cost ($/hr) | Avg $/MWh | Time |
|---|---|---|---|
| Click Solve & Compare All Algorithms | |||
The Algorithms
Three approaches to economic dispatch
Lambda Iteration (Equal Incremental Cost)
O(n log n) | Exact for linear costsThe classical method exploits the optimality condition that all dispatched generators should operate at the same incremental cost λ. Sort generators by marginal cost, then increase λ from the cheapest unit upward, loading each generator to its maximum before moving to the next, until demand is met. For linear cost curves this yields the exact LP optimum.
LP Simplex
O(n) average | Polynomial worst caseFormulate economic dispatch as a standard linear program and solve via the simplex method. The LP has n decision variables (one per generator), one equality constraint (demand balance), and 2n bound constraints. Simplex pivots efficiently through basic feasible solutions. For this small problem structure, simplex typically terminates in a few pivots.
Greedy Merit Order
O(n log n)Sort generators by ascending marginal cost (the merit order). Greedily load each generator to its maximum capacity before moving to the next more expensive unit. The last generator loaded may be partially dispatched to exactly meet demand. This simple approach is used for quick estimates in real-time market operations and produces optimal solutions when all generators have zero minimum output.
Real-World Complexity
Why practical dispatch goes beyond the basic LP model
Beyond Linear Costs
- Quadratic cost curves — Real fuel cost functions are typically quadratic (C = a + b·P + c·P²), requiring QP solvers
- Ramp rate limits — Generators cannot change output instantaneously; ramp constraints couple successive dispatch intervals
- Transmission constraints — Power flow on transmission lines limits how much generation can reach each load zone
- Reserve requirements — A fraction of capacity must be held back for frequency regulation and contingency reserves
- Emission limits — SO2, NOx, and CO2 caps add environmental constraints to the optimization
- Stochastic renewables — Wind and solar forecasts are uncertain; robust or stochastic dispatch is needed
- Unit commitment coupling — Startup/shutdown costs and minimum up/down times link dispatch to the integer commitment problem
Key References
Foundational works in power system optimization
- (2001). “The Next Generation of Electric Power Unit Commitment Models.” Springer. doi:10.1007/978-1-4757-3427-7
- (2014). “Power Generation, Operation, and Control.” 3rd ed. Wiley.
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