Single-Period Economic Dispatch
Economic Dispatch
Decision Problem: Given a fleet of (online) generators, how do we meet demand at lowest cost?
New Constraints: Power plants are generally subject to engineering constraints that we had previously neglected:
- Ramping limits
- Minimum/Maximum power outputs
- May include network constraints (we will ignore here)
Single-Period Economic Dispatch
What are our variables?
| \(d\) |
demand (MW) |
| \(y_g\) |
generation (MW) by generator \(g \in \mathcal{G}\) |
| \(VarCost_g\) |
variable generation cost ($/MWh) for generator \(g\) |
| \(P^{\text{min/max}}_g\) |
generation limits (MW) for generator \(g\) |
Note on Variable Costs
In practice, variable costs come from:
- labor costs;
- equipment upkeep;
- fuel costs (the big one!).
Fuel is the big variable. The translation of fuel costs to generation costs depends on the efficiency (the heat rate) of the plant.
Note on Variable Costs
These costs are often actually quadratic, not linear, due to efficiency changes.
But we can assume a piecewise linear approximation.
Single-Period Economic Dispatch
Then the economic dispatch problem becomes:
\[
\begin{align}
\min_{y_g} & \sum_g VarCost_g \times y_g & \\
\text{subject to:} \quad & \sum_g y_g \geq d & \forall g \in \mathcal{G} \\[0.5em]
& y_g \leq P^{\text{max}}_g & \forall g \in \mathcal{G} \\[0.5em]
& y_g \geq P^{\text{min}}_g & \forall g \in \mathcal{G}
\end{align}
\]
Single-Period Example: Data
- 1 biomass, 50 MW capacity, $5/MWh
- 1 hydroelectric, 500 MW capacity, $0/MWh
- 5 natural gas CCGT, 25-220 MW minimum, 50-620 MW capacity $22-37/MWh
- 6 natural gas CT, 0-73 MW minimum, 48-100 MW capacity, $38-45/MWh
Let’s assume demand is 2400 MW.
Single-Period Results
The cost of operating the system is $51,016, and the shadow price of the demand constraint is -$36.
Cost of Marginal Generation
How can we understand these results?
Dispatch Stack and Merit Order
This supply curve (the dispatch stack) gives the merit order.
Dispatch Stack and Merit Order
What might complicate this simple merit ordering based on variable costs?
Ramping Constraints
Now, let’s consider multiple time periods.
Not only do we need to meet demand at every time period, but we have additional ramping constraints.
Plants can only increase and decrease their output by so much from time to time, by \(R_g\).
Multi-Period Generator Data
Ramping constraints can vary strongly by generator type, which, combined with costs, influences whether we view generators as base load or peaking.
- Nuclear plants generally have a very narrow range in which they can operate;
- Combustion turbine gas plants can ramp from 0-100% very rapidly.
Multi-Period Generator Data
We’ll make this simple for this problem:
- Biomass, hydroelectric, CT can ramp from 0-100% each hour.
- CCGT plants can ramp from 50-100% of maximum capacity.
The “Duck Curve”
Ramping and minimum generation play a major role in systems with high levels of renewable penetration.
For example, a prominent feature of grids with large solar generation is the “duck curve” (right).
