[Paper Review] Simultaneous optimisation of temperature and energy in linear energy system models

Patrik Schönfeldt, Adrian Grimm+7 more|arXiv (Cornell University)|Dec 23, 2020

Integrated Energy Systems Optimization17 references4 citations

TL;DR

This paper proposes a linear programming method for simultaneous optimization of temperature and energy in heat supply systems by discretizing temperature levels, enabling efficient unit commitment and operational temperature selection without integer variables. The approach yields meaningful trade-offs between cost and exergy efficiency, with key results showing 4.7% higher pumping electricity use in exergy-optimized operation due to preference for lower-temperature solar heat.

ABSTRACT

Linear programming is used as a standard tool for optimising unit commitment or power flows in energy supply systems. For heat supply systems, however, it faces a relevant limitation: For them, energy yield depends on the output temperature, thus both quantities would have to be optimised simultaneously and the resulting problem is quadratic. As a solution, we describe a method working with discrete temperature levels. This paper presents mathematical models of various technologies and displays their potential in a case study focused on integrated residential heat and electricity supply. It is shown that the technique yields reasonable results including the choice of operational temperatures.

Motivation & Objective

• Address the limitation of linear programming in simultaneously optimizing temperature and energy in heat systems, which leads to non-linear, quadratic problems.
• Overcome the challenge of non-linear coupling between temperature and heat flow in energy system models, particularly for renewable heat sources like heat pumps and solar thermal.
• Enable practical unit commitment and operational temperature selection in integrated residential energy systems using standard linear solvers.
• Demonstrate the method’s effectiveness through a case study on a solar-based residential energy system with both economic and exergy-based optimization objectives.
• Show that discrete temperature levels can approximate continuous temperature optimization while maintaining computational tractability and physical consistency.

Proposed method

• Discretize the continuous temperature range into k fixed levels, transforming the non-linear problem into a linear one by treating temperature differences as constants per level.
• Model heat flows using the equation $\dot{Q} = \sum_{n=0}^{k-1} \rho c_p d v_n (T_n - T_{\text{low}}) $, where $v_n$ is the velocity at temperature level $T_n$, enabling linear optimization.
• Implement temperature-dependent efficiency and heat transfer using a recursive formulation: $\dot{Q}_{\text{in},n} = (1 - r_{n,n-1}) \dot{Q}_{\text{out},n-1} + r_{n,n-1} \dot{Q}_{s,n}$, with $r_{n,n-1} = \frac{T_n - T_{n-1}}{T_n - T_{\text{low}}}$, to ensure thermodynamic consistency.
• Model heat sources such as heat pumps and solar thermal plants with temperature-dependent efficiencies by defining separate flows at each discrete level.
• Use the open-source energy system modeling framework oemof.solph to implement and validate the model in a real-world residential energy system case study.
• Apply two distinct optimization objectives: minimizing annual costs and minimizing exergy destruction, allowing comparison of operational strategies.

Experimental results

Research questions

• RQ1 Can linear programming be effectively adapted to simultaneously optimize temperature and energy in heat supply systems without introducing non-linear or mixed-integer variables?
• RQ2 How do discrete temperature levels compare to continuous temperature optimization in terms of accuracy and computational efficiency for energy system modeling?
• RQ3 What are the differences in operational strategies—particularly in heat pump use, storage charging, and heating rod timing—when optimizing for economic cost versus exergy efficiency?
• RQ4 To what extent does the choice of operational temperature level (e.g., 30°C vs. 45°C) affect system performance, especially in terms of pumping electricity and solar heat utilization?
• RQ5 How does the method preserve thermodynamic consistency while enabling linear optimization in complex, multi-level heat networks?

Key findings

• The method successfully enables simultaneous optimization of temperature and energy in linear energy system models using discrete temperature levels, avoiding the need for non-linear or mixed-integer programming.
• In the price-optimized case, the heating rod is primarily used at night to leverage lower electricity prices, while in the exergy-optimized case, it is used during the day to align with higher renewable energy availability.
• The exergy-optimized solution prefers the 30°C temperature level for solar heat, resulting in 75.1 MWh/year of solar heat at that level, compared to 14.9 MWh/year in the price-optimized case.
• The price-optimized solution favors the 45°C level for solar heat, using 270 MWh/year of heat at that temperature, compared to 210 MWh/year in the exergy-optimized case.
• The exergy-optimized case requires 4.7% more pumping electricity (11.1 MWh/year vs. 10.6 MWh/year), reflecting the trade-off between exergy efficiency and operational energy use.
• Despite different optimization goals, the overall system behavior and total costs/exergy values differ by only 1%, indicating limited flexibility in the system design but clear operational distinctions in timing and temperature choice.