UnitCommitment.jl/docs/model.md

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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].

Decision variables

Generators

Name	Symbol	Description	Unit
is_on[g,t]	u_{g}(t)	True if generator g is on at time t.	Binary
switch_on[g,t]	v_{g}(t)	True is generator g switches on at time t.	Binary
switch_off[g,t]	w_{g}(t)	True if generator g switches off at time t.	Binary
prod_above[g,t]	p'_{g}(t)	Amount of power produced by generator g above its minimum power output at time t. For example, if the minimum power of generator g is 100 MW and g is producing 115 MW of power at time t, then prod_above[g,t] equals 15.0.	MW
segprod[g,t,k]	p^k_g(t)	Amount of power from piecewise linear segment k produced by generator g at time t. For example, if cost curve for generator g is defined by the points (100, 1400), (110, 1600), (130, 2200) and (135, 2400), and if the generator is producing 115 MW of power at time t, then segprod[g,t,:] equals [10.0, 5.0, 0.0].	MW
reserve[g,t]	r_g(t)	Amount of reserves provided by generator g at time t.	MW
startup[g,t,s]	\delta^s_g(t)	True if generator g switches on at time t incurring start-up costs from start-up category s.	Binary

Buses

Name	Symbol	Description	Unit
net_injection[b,t]	n_b(t)	Net injection at bus b at time t.	MW
curtail[b,t]	s^+_b(t)	Amount of load curtailed at bus b at time t	MW

Price-sensitive loads

Name	Symbol	Description	Unit
loads[s,t]	d_{s}(t)	Amount of power served to price-sensitive load s at time t.	MW

Transmission lines

Name	Symbol	Description	Unit
flow[l,t]	f_l(t)	Power flow on line l at time t.	MW
overflow[l,t]	f^+_l(t)	Amount of flow above the limit for line l at time t.	MW

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.

Objective function

\begin{align}
    \text{minimize} \;\; &
        \sum_{t \in \mathcal{T}}
          \sum_{g \in \mathcal{G}}
          C^\text{min}_g(t) u_g(t) \\
    &
        + \sum_{t \in \mathcal{T}}
          \sum_{g \in \mathcal{G}}
          \sum_{g \in \mathcal{K}_g}
          C^k_g(t) p^k_g(t) \\
    &
        + \sum_{t \in \mathcal{T}}
          \sum_{g \in \mathcal{G}}
          \sum_{s \in \mathcal{S}_g}
          C^s_{g}(t) \delta^s_g(t) \\
    &
        + \sum_{t \in \mathcal{T}}
          \sum_{l \in \mathcal{L}}
          C^\text{overflow}_{l}(t) f^+_l(t) \\
    &
        + \sum_{t \in \mathcal{T}}
           \sum_{b \in \mathcal{B}}
           C^\text{curtail}(t) s^+_b(t) \\
    &
        - \sum_{t \in \mathcal{T}}
           \sum_{s \in \mathcal{PS}}
           R_{s}(t) d_{s}(t) \\
          
\end{align}

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)

Constraints

TODO

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
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)

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