Dispatchable/Non-dispatchable generators

Overview

Electric power systems rely on a mix of generation technologies. Broadly, these fall into two categories:

• Dispatchable generators: Facilities whose output can be adjusted up or down by the system operator (e.g., natural gas, coal), subject to physical limits (ramping limits, minimum up/down times).

• Non‑dispatchable generators: Units that produce energy according to an exogenous resource profile (e.g., wind turbines, solar PV), with limited or no ability for the operator to increase output beyond what nature provides.

Dispatchable generators

We model the operation of dispatchable generators using three variables:

• Dispatch variable \(p_{g,t}\): Power output of generator \(g\) at time \(t\). More specifically, it is split into “above-min” generation \(p'_{g,t}\) and “at-min” generation \(\underline{P}_g\)

• Commitment variable \(u_{g,t} \in \{0,1\}\): Binary indicator if \(g\) is online.

• Starting/Shutdown variable \(v_{g,t}, w_{g,t} \in \{0,1\}\): Binary indicator if \(g\) is starting up or shutting down at time \(t\).

Each dispatchable generator is subject to constraints:

• Capacity: The capacity is constrainted by the minimum and maximum capacity.

  \(\underline{P}_g \times u_{g,t} \le P'_{g,t} + \underline{P}_g \le \bar{P}_g \times u_{g,t}\)

• Ramping limit: The change in power output from \(t\) to \(t+1\) cannot be over the ramping limit.

• Minimum up/down time: Once started or stopped, the unit remains in that state for the specified duration.

• Must take: If the unit has to be included in the generation profile.

We then minimize the cost of generation, for which the cost of individual dispatchable generator is given by:

• Fixed cost: Fixed cost is a function of rated capacity and fixed cost per unit.

  \(c_{g,t}^{fixed} = \bar{P}_g \times {fixed\_cost\_per\_unit}_g \times u_{g,t}\)

• Variable cost: Variable cost is a function of fuel cost, heat rate, and operating cost.

  \(c_{g,t}^{var} = (({fuel\_price}_g \times {heat\_rate}_g) + {opex}_g) \times p_{g,t}\)

• Startup cost: Startup cost is a function of rated capacity and startup cost per unit.

  \(c_{g,t}^{start} = P_g^{max} \times {startup\_cost}_g \times v_{g,t}\)

• Curtailment cost: Curtailing “must-take” thermal output is priced as the same variable rate.

  \(c_{g,t}^{curt} = (({fuel\_price}_g \times {heat\_rate}_g) + {opex}_g) \times p^curt_{g,t}\)

Non-dispatchable generators

For non-dispatchable generators, the model’s decision is to dispatch, curtail, or store the renewable energy produced. We therefore have:

• Dispatched \(pdispatch_{g,t}\)

• Curtailed \(pcurtail_{g,t}\)

• Charged \(pcharge_{g,t}\)

Non-dispatchable generators are subject to constraints:

• Available capacity:

  \(pdispatch_{g,t} \le available\_capacity_{g,t}\)

• Energy balance:

  \(pdispatch_{g,t} + pcurtail_{g,t} + pcharge_{g,t} = available\_capacity_{g,t}\)

We assume non-dispatchable generators do not have a fixed or start-up cost, and we get the variable cost from the contract price:

• Variable cost:

  \(c_{g,t}^{var} = {contract\_price}_g \times pdispatch_{g,t}\)