Lecture VI - Unit Commitment Problem
Energy System Optimization with Julia
Quick Recap from last Week
NamedTuples in Julia
- NamedTuples are immutable collections of key-value pairs
- They provide a convenient way to group related data
- Can be created using a function that returns a tuple with named fields
# Example: Creating a NamedTuple for a student
function Student(name::String, age::Int, grade::Float64)
return (name = name, age = age, grade = grade)
end
# Create a dictionary of students
students = Dict(
"Alice" => Student("Alice", 20, 3.7),
"Bob" => Student("Bob", 21, 3.5)
)
# Access data using dot notation
println(students["Alice"].grade) # Output: 3.7Model-Solving Function Pattern
- Encapsulate optimization models in functions for reusability
- Return results as NamedTuples for easy access
- Store results in DataFrames for analysis
# Example data for different scenarios
data = Dict(
"base" => Dict(
"cost" => 10.0,
"limit" => 100.0
),
"high_cost" => Dict(
"cost" => 15.0,
"limit" => 100.0
),
"low_limit" => Dict(
"cost" => 10.0,
"limit" => 50.0
)
)
# Example: Function to solve a simple optimization problem
function solve_optimization(data::Dict)
model = Model(HiGHS.Optimizer)
set_silent(model)
# Define variables and solve model
@variable(model, x >= 0)
@objective(model, Min, data["cost"] * x)
@constraint(model, x <= data["limit"])
optimize!(model)
# Return results as NamedTuple
return (
solution = value(x),
objective = objective_value(model)
)
end
# Store results in DataFrame
results = DataFrame(
scenario = String[],
solution = Float64[],
objective = Float64[]
)
# Solve multiple scenarios
for (name, scenario_data) in data
solution = solve_optimization(scenario_data)
push!(results, (name, solution.solution, solution.objective))
endEconomic Dispatch Problem Recap
- Objective: Minimize total generation cost
- Decision Variables:
- Power output of thermal generators (\(p_g\))
- Power output of wind turbines (\(p_w\))
- Key Constraints:
- Power balance: \(\sum_g p_g + \sum_w p_w = d^f\)
- Generator limits: \(p^{\min}_g \leq p_g \leq p^{\max}_g\)
- Wind limits: \(0 \leq p_w \leq p^f_w\)
The Economic Dispatch problem optimizes power generation for a single time step, assuming all generators are already committed to operation.
From Economic Dispatch to Unit Commitment
Limitations of Economic Dispatch: - Assumes generators are already running - Ignores start-up and shut-down costs - Doesn’t consider minimum up/down times - Single time step optimization
Unit Commitment Solution: - Adds binary variables for generator on/off status - Considers multiple time steps - Includes start-up/shut-down costs - Enforces minimum up/down time, ramping, etc. constraints - More realistic but computationally more complex
Unit Commitment extends Economic Dispatch by adding operational constraints and time-dependent decisions.
Solutions from last Week
- The tutorials from last week will again be available on Friday
- You can access them in the project folder on Github
- Click on the little cat icon on the bottom right
. . .
You can ask questions anytime in class or via email!
Introduction
Power System Basic Models: Unit Commitment
Power system with generators and an aggregated demand
- Generators can be switched on/off to meet the demand
- Generators have different costs and operational constraints
- Decisions are made over multiple time steps
Unit Commitment Problem
- Objective: Minimize the total cost of producing electricity
- Constraints:
- Demand must be met
- Generators have limits on their power output
- Renewable power injection
- Generator on/off status
- Minimum up/down times
- Ramp rate limits
- Decision variables:
- Power output of thermal generators
- Power output of renewables
- Binary variables for generator status
Problem Structure
Available Sets
Question: What are the sets we need?
. . .
- \(\mathcal{G}\) - Set of thermal generators indexed by \(g \in \{1,2,...,|\mathcal{G}|\}\)
- \(\mathcal{W}\) - Set of wind turbines indexed by \(w \in \{1,2,...,|\mathcal{W}|\}\)
- \(\mathcal{T}\) - Set of time periods indexed by \(t \in \{1,2,...,|\mathcal{T}|\}\)
Available Parameters
Question: What are possible parameters?
. . .
- \(C^{var}_g\) - Variable cost of thermal generator \(g\in\mathcal{G}\) in [\(EUR/\text{MWh}\)]
- \(C^{fix}_g\) - Fixed cost of thermal generator \(g\in\mathcal{G}\) in [\(EUR\)]
- \(C^{var}_w\) - Cost of wind turbine \(w\in\mathcal{W}\) in [\(EUR/\text{MWh}\)]
- \(P^{\min}_g\) - Minimum power output of thermal generator \(g\in\mathcal{G}\) in [MW]
- \(P^{\max}_g\) - Maximum power output of thermal generator \(g\in\mathcal{G}\) in [MW]
- \(P^{f}_{w,t}\) - Forecasted power output of wind turbine \(w\in\mathcal{W}\) at time \(t\) in [MW]
- \(D^f_t\) - Forecasted demand at time \(t\) in [MW]
- \(T^{up}_g\) - Minimum up time of generator \(g\in\mathcal{G}\) in [h]
- \(T^{down}_g\) - Minimum down time of generator \(g\in\mathcal{G}\) in [h]
- \(R^{up}_g\) - Maximum ramp-up rate of generator \(g\in\mathcal{G}\) in [MW/h]
- \(R^{down}_g\) - Maximum ramp-down rate of generator \(g\in\mathcal{G}\) in [MW/h]
Decision Variables
- \(p_{g,t}\) - Power output of thermal generator \(g\in\mathcal{G}\) at time \(t\)
- \(p_{w,t}\) - Power injection of wind turbine \(w\in\mathcal{W}\) at time \(t\)
- \(u_{g,t}\) - Binary variable indicating if generator \(g\in\mathcal{G}\) is on at time \(t\)
Model Formulation
Objective Function
\(\text{Minimize} \quad \sum_{t \in \mathcal{T}} \left( \sum_{g \in \mathcal{G}} (C^{var}_g p_{g,t} + C^{fix}_g u_{g,t}) + \sum_{w \in \mathcal{W}} C^{var}_w p_{w,t} \right)\)
- Variable costs for power generation (\(C^{var}_g p_{g,t}\)): Cost per hour of electricity produced
- Fixed costs for keeping generators on (\(C^{fix}_g u_{g,t}\)): Cost of maintaining a generator in operation
- Wind power costs (\(C^{var}_w p_{w,t}\)): Variable costs per hour for wind power production
Power Balance Constraints
\(\sum_{g \in \mathcal{G}} p_{g,t} + \sum_{w \in \mathcal{W}} p_{w,t} = D^f_t \quad \forall t \in \mathcal{T}\)
This constraint ensures that at each time step \(t\): - The sum of all thermal generator outputs (\(\sum_{g \in \mathcal{G}} p_{g,t}\)) - Plus all wind power outputs (\(\sum_{w \in \mathcal{W}} p_{w,t}\)) - Must exactly match the forecasted demand (\(D^f_t\))
Generator Limits Constraints
\(P^{\min}_g u_{g,t} \leq p_{g,t} \leq P^{\max}_g u_{g,t} \quad \forall g \in \mathcal{G}, t \in \mathcal{T}\)
For each generator \(g\) at time \(t\): - If the generator is on (\(u_{g,t} = 1\)), its output must be between minimum (\(P^{\min}_g\)) and maximum (\(P^{\max}_g\)) power - If the generator is off (\(u_{g,t} = 0\)), its output must be zero - The binary variable \(u_{g,t}\) ensures these limits are enforced
Wind Power Injection Constraints
\(0 \leq p_{w,t} \leq P^f_{w,t} \quad \forall w \in \mathcal{W}, t \in \mathcal{T}\)
For each wind turbine \(w\) at time \(t\): - Power output must be non-negative - Cannot exceed the forecasted available wind power (\(P^f_{w,t}\)) - Unlike thermal generators, wind turbines don’t have binary variables as the power injection is controllable between 0 and the forecasted available wind power
Minimum Up/Down Time Constraints
\(u_{g,t} - u_{g,t-1} \leq u_{g,\tau} \quad \forall g \in \mathcal{G}, t \in \mathcal{T}, \tau \in [t+1, \min(t+T^{up}_g-1,|\mathcal{T}|)]\)
\(u_{g,t-1} - u_{g,t} \leq 1 - u_{g,\tau} \quad \forall g \in \mathcal{G}, t \in \mathcal{T}, \tau \in [t+1, \min(t+T^{down}_g-1,|\mathcal{T}|)]\)
These constraints enforce minimum operating times: - First equation: If a generator starts up at time \(t\), it must stay on for at least \(T^{up}_g\) time periods - Second equation: If a generator shuts down at time \(t\), it must stay off for at least \(T^{down}_g\) time periods - \(\tau\) represents the time periods during which these constraints must be enforced
Ramp Rate Constraints
\(p_{g,t} - p_{g,t-1} \leq R^{up}_g \quad \forall g \in \mathcal{G}, t \in \mathcal{T}\)
\(p_{g,t-1} - p_{g,t} \leq R^{down}_g \quad \forall g \in \mathcal{G}, t \in \mathcal{T}\)
These constraints limit how quickly generators can change their output: - First equation: Power increase between consecutive time steps cannot exceed ramp-up rate \(R^{up}_g\) - Second equation: Power decrease between consecutive time steps cannot exceed ramp-down rate \(R^{down}_g\) - Prevents sudden changes in generator output that could damage equipment
Complete Unit Commitment Model
\(\text{Minimize} \quad \sum_{t \in \mathcal{T}} \left( \sum_{g \in \mathcal{G}} (C^{var}_g p_{g,t} + C^{fix}_g u_{g,t}) + \sum_{w \in \mathcal{W}} C^{var}_w p_{w,t} \right)\)
subject to
\(\sum_{g \in \mathcal{G}} p_{g,t} + \sum_{w \in \mathcal{W}} p_{w,t} = D^f_t \quad \forall t \in \mathcal{T}\)
\(P^{\min}_g u_{g,t} \leq p_{g,t} \leq P^{\max}_g u_{g,t} \quad \forall g \in \mathcal{G}, t \in \mathcal{T}\)
\(0 \leq p_{w,t} \leq P^f_{w,t} \quad \forall w \in \mathcal{W}, t \in \mathcal{T}\)
\(u_{g,t} - u_{g,t-1} \leq u_{g,\tau} \quad \forall g \in \mathcal{G}, t \in \mathcal{T}, \tau \in [t+1, \min(t+T^{up}_g-1,|\mathcal{T}|)]\)
\(u_{g,t-1} - u_{g,t} \leq 1 - u_{g,\tau} \quad \forall g \in \mathcal{G}, t \in \mathcal{T}, \tau \in [t+1, \min(t+T^{down}_g-1,|\mathcal{T}|)]\)
\(p_{g,t} - p_{g,t-1} \leq R^{up}_g \quad \forall g \in \mathcal{G}, t \in \mathcal{T}\)
\(p_{g,t-1} - p_{g,t} \leq R^{down}_g \quad \forall g \in \mathcal{G}, t \in \mathcal{T}\)
\(u_{g,t} \in \{0,1\} \quad \forall g \in \mathcal{G}, t \in \mathcal{T}\)
Model Characteristics
Model Type
Question: What type of optimization problem is this?
. . .
- Mixed-Integer Linear Programming (MILP)
- Binary variables for generator status
- Linear constraints and objective
- Time-dependent decisions
Solution Methods
Question: How can we solve this problem?
. . .
- Branch and Bound algorithm
- Commercial solvers (Gurobi, CPLEX)
- Open-source solvers (HiGHS)
- Decomposition methods for large problems
Computational Challenges
Question: What makes this problem challenging?
. . .
- Large number of binary variables
- Time-dependent inter-temporal constraints
- Minimum up/down time constraints
- Ramp rate constraints
- Multiple time periods
Impact
Real-World Applications
- Generation scheduling
- Day-ahead market clearing
- Ancillary services
- Reserve capacity planning
Further Extensions
- Battery storage
- Demand response
- Network constraints
- Reserve requirements
- Multiple fuel types
- Emission constraints
- Uncertainty in forecasts
We have covered the Unit Commitment problem and its mathematical formulation. The tutorial will help you implement and solve this problem using Julia and JuMP.
Literature
Literature I
For more interesting literature to learn more about Julia, take a look at the literature list of this course.
Literature II
For a detailed mathematical formulation of the Unit Commitment problem, see Morales-Espana, Latorre, and Ramos (2013) and Zimmermann and Kather (2019).