4. JuMP Model¶
+In this page, we describe the JuMP optimization model produced by the function UnitCommitment.build_model. A detailed understanding of this model is not necessary if you are just interested in using the package to solve some standard unit commitment cases, but it may be useful, for example, if you need to solve a slightly different problem, with additional variables and constraints. The notation in this page generally follows [KnOsWa20].
4.1. Decision variables¶
+Generators¶
+Name |
+Symbol |
+Description |
+Unit |
+
|---|---|---|---|
|
+\(u_{g}(t)\) |
+True if generator |
+Binary |
+
|
+\(v_{g}(t)\) |
+True is generator |
+Binary |
+
|
+\(w_{g}(t)\) |
+True if generator |
+Binary |
+
|
+\(p'_{g}(t)\) |
+Amount of power produced by generator |
+MW |
+
|
+\(p^k_g(t)\) |
+Amount of power from piecewise linear segment |
+MW |
+
|
+\(r_g(t)\) |
+Amount of reserves provided by generator |
+MW |
+
|
+\(\delta^s_g(t)\) |
+True if generator |
+Binary |
+
Buses¶
+Name |
+Symbol |
+Description |
+Unit |
+
|---|---|---|---|
|
+\(n_b(t)\) |
+Net injection at bus |
+MW |
+
|
+\(s^+_b(t)\) |
+Amount of load curtailed at bus |
+MW |
+
Price-sensitive loads¶
+Name |
+Symbol |
+Description |
+Unit |
+
|---|---|---|---|
|
+\(d_{s}(t)\) |
+Amount of power served to price-sensitive load |
+MW |
+
Transmission lines¶
+Name |
+Symbol |
+Description |
+Unit |
+
|---|---|---|---|
|
+\(f_l(t)\) |
+Power flow on line |
+MW |
+
|
+\(f^+_l(t)\) |
+Amount of flow above the limit for line |
+MW |
+
Danger
+Since transmission and N-1 security constraints are enforced in a lazy way, most of the variables flow[l,t] and overflow[l,t] are never added to the model. Accessing model[:flow][l,t], for example, without first checking that the variable exists will likely generate an error.
4.2. Objective function¶
+where
+-
+
\(\mathcal{B}\) is the set of buses
+\(\mathcal{G}\) is the set of generators
+\(\mathcal{L}\) is the set of transmission lines
+\(\mathcal{PS}\) is the set of price-sensitive loads
+\(\mathcal{S}_g\) is the set of start-up categories for generator \(g\)
+\(\mathcal{T}\) is the set of time steps
+\(C^\text{curtail}(t)\) is the curtailment penalty (in $/MW)
+\(C^\text{min}_g(t)\) is the cost of keeping generator \(g\) on and producing at minimum power during time \(t\) (in $)
+\(C^\text{overflow}_{l}(t)\) is the flow limit penalty for line \(l\) at time \(t\) (in $/MW)
+\(C^k_g(t)\) is the cost for generator \(g\) to produce 1 MW of power at time \(t\) under piecewise linear segment \(k\)
+\(C^s_{g}(t)\) is the cost of starting up generator \(g\) at time \(t\) under start-up category \(s\) (in $)
+\(R_{s}(t)\) is the revenue obtained from serving price-sensitive load \(s\) at time \(t\) (in $/MW)
+
4.3. Constraints¶
+TODO
+4.4. Inspecting and modifying the model¶
+Accessing decision variables¶
+After building a model using UnitCommitment.build_model, it is possible to obtain a reference to the decision variables by calling model[:varname][index]. For example, model[:is_on]["g1",1] returns a direct reference to the JuMP variable indicating whether generator named “g1” is on at time 1. The script below illustrates how to build a model, solve it and display the solution without using the function UnitCommitment.solution.
using Cbc
+using Printf
+using JuMP
+using UnitCommitment
+
+# Load benchmark instance
+instance = UnitCommitment.read_benchmark("matpower/case118/2017-02-01")
+
+# Build JuMP model
+model = UnitCommitment.build_model(
+ instance=instance,
+ optimizer=Cbc.Optimizer,
+)
+
+# Solve the model
+UnitCommitment.optimize!(model)
+
+# Display commitment status
+for g in instance.units
+ for t in 1:instance.time
+ @printf(
+ "%-10s %5d %5.1f %5.1f %5.1f\n",
+ g.name,
+ t,
+ value(model[:is_on][g.name, t]),
+ value(model[:switch_on][g.name, t]),
+ value(model[:switch_off][g.name, t]),
+ )
+ end
+end
+Modifying the model¶
+Since we now have a direct reference to the JuMP decision variables, it is possible to fix variables, change the coefficients in the objective function, or even add new constraints to the model before solving it. The script below shows how can this be accomplished. For more information on modifying an existing model, see the JuMP documentation.
+using Cbc
+using JuMP
+using UnitCommitment
+
+# Load benchmark instance
+instance = UnitCommitment.read_benchmark("matpower/case118/2017-02-01")
+
+# Construct JuMP
+model = UnitCommitment.build_model(
+ instance=instance,
+ optimizer=Cbc.Optimizer,
+)
+
+# Fix a decision variable to 1.0
+JuMP.fix(
+ model[:is_on]["g1",1],
+ 1.0,
+ force=true,
+)
+
+# Change the objective function
+JuMP.set_objective_coefficient(
+ model,
+ model[:switch_on]["g2",1],
+ 1000.0,
+)
+
+# Create a new constraint
+@constraint(
+ model,
+ model[:is_on]["g3",1] + model[:is_on]["g4",1] <= 1,
+)
+
+# Solve the model
+UnitCommitment.optimize!(model)
+Adding components to a bus¶
+4.5. References¶
+-
+
[KnOsWa20] Bernard Knueven, James Ostrowski and Jean-Paul Watson. “On Mixed-Integer Programming Formulations for the Unit Commitment Problem”. INFORMS Journal on Computing (2020). DOI: 10.1287/ijoc.2019.0944
+