[Paper Review] Energy System Optimisation using (Mixed Integer) Linear Programming

Sebastian Miehling, Andreas Hänel+3 more|arXiv (Cornell University)|Aug 3, 2023

Integrated Energy Systems Optimization4 citations

TL;DR

This paper presents a comprehensive framework for energy system optimization using mixed integer linear programming (MILP), demonstrating how to model energy systems as networks of nodes and edges with linear constraints. It details the mathematical formulation of cost-minimizing dispatch and capacity expansion, including advanced features like discrete units, minimum up/down times, and storage, with a practical example showing optimal capacity and scheduling for a multi-energy system over 8,760 hourly time steps.

ABSTRACT

Although energy system optimisation based on linear optimisation is often used for influential energy outlooks and studies for political decision-makers, the underlying background still needs to be described in the scientific literature in a concise and general form. This study presents the main equations and advanced ideas and explains further possibilities mixed integer linear programming offers in energy system optimisation. Furthermore, the equations are shown using an example system to present a more practical point of view. Therefore, this study is aimed at researchers trying to understand the background of studies using energy system optimisation and researchers building their implementation into a new framework. This study describes how to build a standard model, how to implement advanced equations using linear programming, and how to implement advanced equations using mixed integer linear programming, as well as shows a small exemplary system. - Presentation of the OpTUMus energy system optimisation framework - Set of equations for a fully functional energy system model - Example of a simple energy system model

Motivation & Objective

• To provide a clear, general, and concise explanation of the mathematical foundations behind energy system optimization using linear and mixed integer linear programming.
• To address the gap in scientific literature where complex models like WEM, TIMES, and REMod are used without sufficient methodological transparency.
• To enable researchers and developers to implement energy system optimization models by offering a standardized framework (OpTUMus) and practical implementation examples.
• To demonstrate how advanced operational constraints—such as minimum up/down times and discrete unit deployment—can be modeled within a MILP framework.
• To present a fully functional, small-scale energy system example with explicit equations and results to serve as a reference for model development and validation.

Proposed method

• Model energy systems as a network of nodes (energy/heat/electricity) and edges (components like PV, CHP, batteries) with conservation-of-flow equations at each node and time step.
• Formulate the objective function as total system cost minimization, including investment, fuel, and operational costs, with time-varying parameters for each component.
• Implement continuous variables for power output and binary variables for on/off states, with constraints ensuring physical and operational limits (e.g., capacity, ramp rates).
• Introduce discrete unit modeling via constraints linking the number of operational units to binary on/off variables, enabling optimization of unit count.
• Apply minimum up/down time constraints using cumulative binary variable sums over time windows, ensuring realistic start-up and shut-down behavior.
• Use the OpTUMus framework to implement and validate the model on a 5-component system (PV, CHP, gas turbine, battery, heat pump) over 8,760 hourly time steps.

Experimental results

Research questions

• RQ1 How can energy system optimization be systematically formulated using linear programming to minimize total system costs?
• RQ2 What are the key mathematical equations and constraints required to model real-world energy system components and their interactions?
• RQ3 How can mixed integer linear programming extend standard linear programming to model discrete decisions such as the number of installed units or on/off states?
• RQ4 What are the practical implications of including minimum up/down time constraints in energy system optimization models?
• RQ5 How can a small, representative energy system be modeled and solved to demonstrate the full workflow from formulation to result interpretation?

Key findings

• The optimal system configuration includes a base-load operation of the CHP and gas turbine, with the heat pump handling fluctuating heat demand.
• PV and battery storage are used to complement base-load generation, with the battery charging during high PV output and discharging during low output.
• The installed capacities from the optimization are: 64.2 MW PV, 4 MW CHP, 9.4 MW gas turbine, 4.3 MW heat pump, and 75 MW battery, as shown in Table 1.
• The objective function in Equation (30) captures the total cost, including fuel, investment, and operational costs, minimized over all 8,760 time steps.
• The inclusion of minimum up/down time constraints (Equations 28 and 29) ensures realistic unit commitment behavior, preventing frequent cycling.
• The study demonstrates that MILP enables modeling of complex operational constraints while maintaining computational tractability for large-scale systems.