Lecture XII - Course Repetition and Summary

Energy System Optimization with Julia

Author
Affiliation

Dr. Tobias Cors

Hamburg University of Applied Science - Summer 2025

Course Overview

What We Have Covered

This course has been organized into three main parts:

  1. Part I: Introduction to Julia and Modelling (Lectures I-IV)
  2. Part II: Energy System Optimization with Basic Models (Lectures V-VIII)
  3. Part III: Energy System Optimization with Advanced Models (Lectures IX-XI)

. . .

Tip

This repetition lecture will help you consolidate your understanding of all key concepts covered throughout the course.

Part I: Introduction to Julia and Modelling

Lecture I: Welcome and Introduction

Course Objectives

  • Understanding and applying optimization to real-world problems
  • Learning to recognize and solve problem structures
  • Energy system optimization applications

Key Topics Covered

  • Course structure and assignments
  • Energy system optimization applications
  • Julia programming language introduction
  • Real-world applications in operations research

Lecture II: First Steps in Julia

Core Programming Concepts

  • Variables: Assignment and data types
  • Vectors: Creation, indexing, and manipulation
  • Comparisons: Boolean operations and conditional logic
  • Loops: for and while loops for iteration
  • Dictionaries: Key-value data structures

Technical Setup

  • Julia installation and configuration
  • VS Code setup with Julia extension
  • IJulia for Jupyter notebooks
  • Assignment submission process

Lecture III: Packages and Data Management

Advanced Julia Concepts

  • Functions: Definition, arguments, and return values
  • Packages: Installation and usage with using and import
  • DataFrames: Tabular data manipulation
  • Input/Output: File reading and writing operations
  • Plotting: Data visualization with Plots.jl

Key Skills Developed

  • Package management with Pkg
  • Data manipulation with DataFrames.jl
  • File I/O operations with CSV.jl
  • Basic plotting and visualization

Lecture IV: Modelling with JuMP

Mathematical Optimization Foundation

  • JuMP Framework: Optimization modeling in Julia
  • Variables: Declaration with bounds and types
  • Constraints: Equality and inequality constraints
  • Objective Functions: Linear and nonlinear objectives
  • Solvers: Integration with various optimization solvers

Modeling Skills

  • Problem formulation in mathematical notation
  • Translation to JuMP code
  • Solver selection and parameter setting
  • Solution analysis and interpretation

Part II: Energy System Optimization with Basic Models

Lecture V: Economic Dispatch Problem

Problem Definition

  • Objective: Minimize electricity generation costs
  • Constraints: Demand satisfaction, generator limits, renewable integration
  • Decision Variables: Power output of thermal generators and renewables

Mathematical Formulation

Sets:

  • \(\mathcal{G}\) - Set of thermal generators
  • \(\mathcal{W}\) - Set of wind turbines

Parameters:

  • \(c^{var}_g\) - Variable cost of generator \(g\)
  • \(p^{\min}_g, p^{\max}_g\) - Power limits of generator \(g\)
  • \(p^{f}_w\) - Forecasted wind power
  • \(d^f\) - Forecasted demand

Decision Variables:

  • \(p_g\) - Power output of generator \(g\)
  • \(p_w\) - Power output of wind turbine \(w\)

Objective:

\(\min \sum_{g \in \mathcal{G}} c^{var}_g p_g + \sum_{w \in \mathcal{W}} c^{var}_w p_w\)

Constraints:

  • Power balance: \(\sum_{g \in \mathcal{G}} p_g + \sum_{w \in \mathcal{W}} p_w = d^f\)
  • Generator limits: \(p^{\min}_g \leq p_g \leq p^{\max}_g \quad \forall g \in \mathcal{G}\)
  • Wind limits: \(0 \leq p_w \leq p^{f}_w \quad \forall w \in \mathcal{W}\)

Lecture VI: Unit Commitment Problem

Problem Extension

  • Binary Variables: Startup/shutdown decisions
  • Time Dimension: Multi-period optimization
  • Additional Constraints: Minimum up/down times, ramp rates

Key Modeling Concepts

  • Binary Variables: \(v_{g,t}\) (startup), \(u_{g,t}\) (commitment)
  • Minimum Up/Down Times: Prevent rapid cycling
  • Ramp Rate Limits: Constrain power change rates
  • Startup Costs: Additional costs for starting generators

Mathematical Extensions

Additional Sets:

  • \(\mathcal{T}\) - Set of time periods

Additional Variables:

  • \(v_{g,t}\) - Binary startup variable
  • \(u_{g,t}\) - Binary commitment variable

Additional Constraints:

  • Minimum up time: \(\sum_{\tau=t}^{t+UT_g-1} u_{g,\tau} \geq UT_g v_{g,t}\)
  • Minimum down time: \(\sum_{\tau=t}^{t+DT_g-1} (1-u_{g,\tau}) \geq DT_g (1-u_{g,t-1})\)
  • Ramp rate: \(p_{g,t} - p_{g,t-1} \leq RampUp_g u_{g,t}\)

Lecture VII: Storage Systems

Storage Modeling Concepts

  • Energy Balance: Storage level evolution over time
  • Power Limits: Charging and discharging rate constraints
  • Efficiency: Energy conversion losses
  • State of Charge: Energy level constraints

Mathematical Formulation

Sets:

  • \(\mathcal{S}\) - Set of storage systems

Variables:

  • \(e_{s,t}\) - Energy level of storage \(s\) at time \(t\)
  • \(p^{ch}_{s,t}\) - Charging power of storage \(s\) at time \(t\)
  • \(p^{dis}_{s,t}\) - Discharging power of storage \(s\) at time \(t\)

Constraints:

  • Energy balance: \(e_{s,t} = e_{s,t-1} + \eta^{ch}_s p^{ch}_{s,t} - \frac{p^{dis}_{s,t}}{\eta^{dis}_s}\)
  • Energy limits: \(E^{\min}_s \leq e_{s,t} \leq E^{\max}_s\)
  • Power limits: \(0 \leq p^{ch}_{s,t} \leq P^{ch,\max}_s\), \(0 \leq p^{dis}_{s,t} \leq P^{dis,\max}_s\)

Lecture VIII: Investment Planning

Long-term Planning Problem

  • Investment Decisions: Capacity expansion decisions
  • Multi-period Optimization: Long-term planning horizon
  • Cost Components: Investment, operational, and maintenance costs

Key Concepts

  • Annualized Costs: Converting investment costs to annual equivalents
  • Capacity Planning: Optimal sizing of generation and storage
  • Economic Analysis: Cost-benefit evaluation of investments
  • System Reliability: Ensuring adequate capacity margins

Part III: Energy System Optimization with Advanced Models

Lecture IX: Multi-Energy Systems

System Integration

  • Multiple Energy Carriers: Electricity and hydrogen as primary example
  • Energy Conversion: Technologies linking different carriers (electrolyzers)
  • System Optimization: Coordinated optimization across carriers

Key Components

  • Electrolyzers: Electricity to hydrogen conversion
  • Energy Storage: Multiple storage types for different carriers
  • Renewable Generation: Wind and PV integration
  • Grid Connection: Market participation for electricity

Mathematical Extensions

Additional Sets:

  • \(\mathcal{S}_{\text{El}}\) - Subset of electricity storage systems
  • \(\mathcal{S}_{\text{H2}}\) - Subset of hydrogen storage systems
  • \(\mathcal{X}\) - Set of electrolyzers

Additional Variables:

  • \(h_{x,t,\omega}\) - Hydrogen production from electrolyzer \(x\)
  • \(p_{x,t,\omega}\) - Power consumption of electrolyzer \(x\)

Additional Constraints:

  • Hydrogen balance: \(\sum_{x \in \mathcal{X}} h_{x,t,\omega} + \sum_{s \in \mathcal{S}_{\text{H2}}} (p^{dis}_{s,t,\omega} - p^{ch}_{s,t,\omega}) = d_{t,\omega}\)
  • Electrolyzer conversion: \(h_{x,t,\omega} = \beta_x p_{x,t,\omega}\)
  • Electricity balance modified: \(\sum_{w \in \mathcal{W}} p_{w,t,\omega} + \sum_{v \in \mathcal{V}} p_{v,t,\omega} + (p^{in}_{t,\omega} - p^{out}_{t,\omega}) + \sum_{s \in \mathcal{S}_{\text{El}}} (p^{dis}_{s,t,\omega} - p^{ch}_{s,t,\omega}) = \sum_{x \in \mathcal{X}} p_{x,t,\omega}\)

Two-Stage Stochastic Programming

  • First Stage (Investment): Storage capacity, wind/PV capacity, electrolyzer capacity decisions
  • Second Stage (Operation): Power dispatch, storage operation, grid interaction decisions
  • Solution Approaches: Monolithic approach or decomposition methods (Benders, Progressive Hedging)
  • Mathematical Form: \(\min_{x} \quad c^T x + \mathbb{E}_{\omega}[Q(x,\omega)]\)

Model Characteristics

  • Two-stage stochastic Linear Programming (LP) problem for multi-energy systems
  • Combines investment and operational decisions across multiple energy carriers
  • Large-scale optimization problem with energy conversion technologies
  • Integrates electricity and hydrogen value chains

Key Extensions from Single-Energy Systems

  • Electrolyzers: Convert electricity to hydrogen with efficiency \(\beta_x\)
  • Hydrogen Storage: Separate storage systems for hydrogen with different characteristics
  • Energy Conversion: Explicit modeling of energy carrier conversion processes
  • Multi-Carrier Storage: Distinguishes between electricity (\(\mathcal{S}_{\text{El}}\)) and hydrogen (\(\mathcal{S}_{\text{H2}}\)) storage
Note

Note: Lecture IX uses technology-specific sets and variables (e.g., \(\mathcal{X}\) for electrolyzers, \(\mathcal{S}_{\text{El}}\) and \(\mathcal{S}_{\text{H2}}\) for storage types). Only the storage modeling provides a template for a more general framework formulation.

Lecture X: Test Exam Preparation

Note

Note: Lecture X was dedicated to test exam preparation, where students practiced exam-style problems together in class. This session focused on applying the concepts learned throughout the course to prepare for the final examination.

Lecture XI: Framework for Multi-Energy System Optimization

General Framework Approach

  • Adaptable Design: Framework that can be configured for different MES setups
  • Configurable Sets and Parameters: Only requires configuration of sets and parameters for new applications
  • Technology-Agnostic: Not limited to specific technologies or energy carriers
  • Scalable Architecture: Can handle various system sizes and complexities

Comprehensive Framework

  • Multi-Carrier Integration: Flexible approach to multiple energy carriers
  • Stochastic Formulation: Uncertainty in renewable generation and demand
  • Investment and Operations: Combined planning and operational optimization

Advanced Modeling Concepts

  • Big M Constraints: Conditional constraint modeling
  • Mixed-Integer Programming: Binary variables for discrete decisions
  • Large-Scale Optimization: Efficient solution methods
  • Scenario Generation: Creating representative uncertainty scenarios

Framework Components

Investment Variables:

  • Storage capacities: \(e^{nom}_s\), \(p^{ch,nom}_s\), \(p^{dis,nom}_s\)
  • Generation capacities: \(p^{nom}_w\), \(p^{nom}_v\)
  • Conversion capacities: \(p^{nom}_x\)

Operational Variables:

  • Power outputs: \(p_{w,t,\omega}\), \(p_{v,t,\omega}\)
  • Storage operation: \(p^{ch}_{s,t,\omega}\), \(p^{dis}_{s,t,\omega}\), \(e_{s,t,\omega}\)
  • Conversion operation: \(p_{x,t,\omega}\), \(h_{x,t,\omega}\)

Two-Stage Stochastic Programming

  • First Stage (Investment): Storage capacity, wind/PV capacity, electrolyzer capacity decisions
  • Second Stage (Operation): Power dispatch, storage operation, grid interaction decisions
  • Solution Approaches: Monolithic approach or decomposition methods (Benders, Progressive Hedging)
  • Mathematical Form: \(\min_{x} \quad c^T x + \mathbb{E}_{\omega}[Q(x,\omega)]\)

Big M Constraints in Mixed-Integer Programming

  • Purpose: Create conditional relationships between continuous and binary variables
  • General Form: \(f(x) \leq M (1 - y)\) where \(M\) is a large constant
  • Application: Modeling complex logical relationships and conditional constraints
  • Example: Generator startup timing constraints in unit commitment problems
Tip

Key Difference: Unlike Lecture IX’s technology-specific approach, Lecture XI introduces a general framework that can be adapted to different MES setups by simply configuring the sets and parameters, making it applicable to various energy system configurations beyond just electricity-hydrogen systems.

Key Mathematical Concepts

Optimization Problem Structure

General Form

\(\min f(x)\)

subject to:

\(g_i(x) \leq 0 \quad \forall i \in \mathcal{I}\) \(h_j(x) = 0 \quad \forall j \in \mathcal{J}\) \(x \in \mathcal{X}\)

Problem Types

  • Linear Programming (LP): Linear objective and constraints
  • Mixed-Integer Linear Programming (MILP): Linear with binary variables
  • Nonlinear Programming (NLP): Nonlinear objective or constraints
  • Stochastic Programming: Multiple scenarios with probabilities

Constraint Types

Equality Constraints

  • Power balance: \(\sum_{g} p_g = d\)
  • Energy balance: \(e_t = e_{t-1} + p^{ch}_t - p^{dis}_t\)

Inequality Constraints

  • Capacity limits: \(p_g \leq p^{\max}_g\)
  • Ramp rates: \(p_{g,t} - p_{g,t-1} \leq RampUp_g\)

Binary Variable Constraints

  • Minimum up time: \(\sum_{\tau=t}^{t+UT-1} u_{\tau} \geq UT \cdot v_t\)
  • Logical constraints: \(p_g \leq p^{\max}_g \cdot u_g\)

Solution Methods

Solver Types

  • LP Solvers: HiGHS, GLPK, Gurobi
  • MILP Solvers: Gurobi, CPLEX, HiGHS
  • NLP Solvers: IPOPT, KNITRO

Solution Analysis

  • Feasibility: Check if solution satisfies all constraints
  • Optimality: Verify solution is globally optimal
  • Sensitivity: Analyze impact of parameter changes

Julia Programming Skills

Core Programming Concepts

Data Types and Structures

# Variables and basic types
x = 5.0          # Float64
y = [1, 2, 3]    # Vector{Int64}
z = Dict("a" => 1, "b" => 2)  # Dict{String, Int64}

# Arrays and matrices
A = [1 2; 3 4]   # Matrix
B = zeros(3, 3)  # 3x3 zero matrix

Control Structures

# Loops
for i in 1:10
    println(i)
end

# Conditionals
if x > 0
    println("Positive")
elseif x < 0
    println("Negative")
else
    println("Zero")
end

Functions

# Function definition
function calculate_cost(power, cost_rate)
    return power * cost_rate
end

# Anonymous functions
f = x -> x^2 + 2*x + 1

Data Management

DataFrames

using DataFrames

# Create DataFrame
df = DataFrame(
    generator = ["G1", "G2", "G3"],
    capacity = [100, 150, 200],
    cost = [50, 60, 45]
)

# Access and manipulate
df.capacity[1] = 120
df[!, :efficiency] = [0.8, 0.85, 0.9]

File I/O

using CSV

# Read CSV file
data = CSV.read("generators.csv", DataFrame)

# Write CSV file
CSV.write("results.csv", results_df)

Optimization with JuMP

Basic Model Structure

using JuMP, HiGHS

# Create model
model = Model(HiGHS.Optimizer)

# Variables
@variable(model, p[g] >= 0)  # Power output
@variable(model, u[g, t], Bin)  # Binary commitment

# Objective
@objective(model, Min, sum(cost[g] * p[g] for g in generators))

# Constraints
@constraint(model, power_balance[t in time], 
    sum(p[g] for g in generators) == demand[t])

# Solve
optimize!(model)

Advanced Modeling

# Conditional constraints with Big M
@constraint(model, p[g] <= max_power[g] * u[g])

# Minimum up time constraints
@constraint(model, min_up_time[g, t], 
    sum(u[g, tau] for tau in t:t+min_up[g]-1) >= min_up[g] * v[g, t])

Energy System Applications

Real-World Problem Types

Generation Planning

  • Economic Dispatch: Optimal power generation for given demand
  • Unit Commitment: Optimal generator scheduling with startup/shutdown
  • Generation Expansion: Long-term capacity planning

Storage Integration

  • Battery Storage: Short-term energy storage
  • Pumped Hydro: Large-scale energy storage
  • Thermal Storage: Heat storage for district heating

Renewable Integration

  • Wind Power: Variable generation with forecasting
  • Solar Power: Diurnal patterns and weather dependence
  • Grid Integration: Balancing variable renewable energy

Multi-Energy Systems

  • Power-to-Gas: Hydrogen production from electricity
  • District Heating: Combined heat and power systems
  • Transport Integration: Electric vehicle charging

Industry Applications

Power System Operations

  • Day-ahead Markets: 24-hour ahead generation scheduling
  • Real-time Balancing: Minute-to-minute power balance
  • Ancillary Services: Frequency regulation and reserves

Energy Planning

  • Investment Planning: Long-term capacity expansion
  • Policy Analysis: Impact of carbon pricing and regulations
  • Technology Assessment: Comparison of different energy technologies

Research Applications

  • Academic Research: Novel optimization algorithms
  • Industry Collaboration: Real-world problem solving
  • Policy Support: Evidence-based energy policy design

Exam Preparation Tips

Key Areas to Review

Mathematical Modeling

  • Problem Formulation: Converting word problems to mathematical models
  • Constraint Types: Understanding different constraint formulations
  • Variable Types: Continuous vs. binary variables
  • Objective Functions: Cost minimization and profit maximization

Julia Programming

  • Basic Syntax: Variables, loops, functions
  • Data Structures: Arrays, DataFrames, Dictionaries
  • JuMP Modeling: Variable declaration, constraints, objectives
  • Solution Analysis: Accessing and interpreting results

Energy System Concepts

  • Power Balance: Supply equals demand constraints
  • Generator Characteristics: Cost curves, capacity limits, ramp rates
  • Storage Operation: Energy balance and efficiency
  • Multi-energy Integration: Energy carrier conversion and storage

Practice Exercises

Modeling Problems

  1. Economic Dispatch: Formulate and solve a simple dispatch problem
  2. Unit Commitment: Add binary variables and startup constraints
  3. Storage Integration: Include storage operation in power balance
  4. Multi-energy: Model electricity-hydrogen system

Programming Problems

  1. Data Manipulation: Read, process, and analyze energy data
  2. Optimization Models: Implement JuMP models for energy problems
  3. Result Analysis: Plot and interpret optimization results
  4. Scenario Analysis: Handle uncertainty in energy systems

Exam Strategy

Time Management

  • Part I (30 points): Energy system modeling (45 minutes)
  • Part II (15 points): Theory and concepts (22 minutes)
  • Part III (15 points): Julia programming (22 minutes)

Problem-Solving Approach

  1. Read Carefully: Understand the problem requirements
  2. Plan Solution: Outline your approach before coding
  3. Test Incrementally: Verify each step before proceeding
  4. Check Results: Ensure your solution makes sense

. . .

Important

Remember: You can bring one handwritten DIN A4 sheet with notes to the exam!

Course Summary

What You Have Learned

Programming Skills

  • Julia Language: Modern, high-performance programming
  • Data Management: Working with real-world energy data
  • Optimization Modeling: Mathematical problem formulation
  • Solution Analysis: Interpreting and visualizing results

Energy System Knowledge

  • Power System Operations: How electricity systems work
  • Optimization Applications: Real-world energy problems
  • Multi-energy Systems: Integration of different energy carriers
  • Uncertainty Handling: Stochastic programming approaches

Problem-Solving Skills

  • Mathematical Modeling: Converting problems to optimization models
  • Algorithm Implementation: Coding optimization algorithms
  • Result Interpretation: Understanding and communicating solutions
  • Team Collaboration: Working in groups on complex problems

Career Applications

Industry Opportunities

  • Energy Companies: Power system planning and operations
  • Consulting Firms: Energy system analysis and optimization
  • Research Institutions: Academic and applied research
  • Government Agencies: Energy policy and regulation

Skill Transfer

  • Programming: Julia skills transfer to other languages
  • Optimization: Mathematical modeling skills for various domains
  • Data Analysis: Energy data skills apply to other sectors
  • Problem Solving: Systematic approach to complex problems

. . .

Tip

The skills you’ve learned in this course are valuable for many careers in energy, operations research, and data science!

Final Thoughts

Course Objectives Achieved

✅ Understanding Optimization: You can now recognize and formulate optimization problems

✅ Julia Programming: You can write, debug, and optimize Julia code

✅ Energy System Modeling: You understand how to model energy system problems

✅ Real-World Applications: You can apply optimization to practical energy problems

✅ Team Collaboration: You’ve worked effectively in groups on complex problems

Next Steps

Further Learning

  • Advanced Optimization: Nonlinear and stochastic programming
  • Machine Learning: Data-driven approaches to energy systems
  • Energy Markets: Market design and price formation
  • Policy Analysis: Impact of regulations on energy systems

Resources

  • Julia Documentation: docs.julialang.org
  • JuMP Documentation: jump.dev
  • Energy Literature: Academic journals and industry reports
  • Online Courses: MOOCs on energy systems and optimization

Stay Connected

  • Research Opportunities: Consider thesis projects in energy optimization
  • Industry Internships: Apply your skills in real-world settings
  • Professional Networks: Join energy and operations research societies
  • Continuing Education: Pursue advanced degrees in related fields

. . .

Note

Thank you for your active participation throughout this course. Your questions and engagement have made this a rewarding experience for everyone!

Literature

Core References

Julia Programming

  • Lauwens, B., & Downey, A. B. (2019). Think Julia: How to think like a computer scientist. O’Reilly®.

Optimization

  • Winston, W. L. (2004). Operations Research: Applications and Algorithms. Thomson Brooks/Cole.
  • Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. McGraw-Hill.

Energy Systems

  • Wood, A. J., & Wollenberg, B. F. (2012). Power Generation, Operation, and Control. Wiley.
  • Kirschen, D. S., & Strbac, G. (2018). Fundamentals of Power System Economics. Wiley.

Multi-Energy Systems

  • Mancarella, P. (2014). MES (multi-energy systems): An overview of concepts and evaluation models. Energy, 65, 1-17.
  • Geidl, M., & Andersson, G. (2007). Optimal power flow of multiple energy carriers. IEEE Transactions on Power Systems, 22(1), 145-155.

Online Resources

Documentation

Tutorials and Examples

Research Papers

  • Recent papers in journals like:
    • IEEE Transactions on Power Systems
    • Energy
    • Applied Energy
    • European Journal of Operational Research