When energy researchers build mathematical models of heat pumps, batteries, and rooftop solar panels, they face a stubborn trade-off: the more detailed the model, the more computationally expensive it becomes, yet oversimplified models can miss the very behavior that matters for controlling thousands of devices on the grid. A new study published in Energy Informatics proposes a way to resolve this dilemma by borrowing a concept from cooperative game theory. Researchers Chang Li, Kevin Förderer, Jörg Matthes, and Veit Hagenmeyer at the Karlsruhe Institute of Technology applied the Shapley value, a method for fairly dividing credit among players in a coalition, to the individual submodels that make up a white-box model of a distributed energy resource. Their goal was to determine, with a single number, how much each submodel contributes to the overall utility of the model.

The Shapley value was introduced by mathematician Lloyd Shapley in 1951 as a solution concept for n-person cooperative games. It quantifies each player’s contribution by evaluating the marginal impact the player makes when added to every possible subset of the other players, then averaging those impacts. Shapley proved in 1953 that this is the only solution satisfying four foundational properties for contribution measures: efficiency, symmetry, linearity, and the null player condition. Efficiency guarantees that the sum of all players’ Shapley values equals the value of the full coalition, so the gain is distributed completely. Symmetry ensures equal contributors receive equal credit, linearity makes the values additive across different value functions, and the null player property assigns zero to anyone who contributes nothing. These properties have carried the method far beyond game theory into explainable artificial intelligence, cost allocation, and energy systems.

In machine learning, the Shapley value underpins SHAP, the SHapley Additive exPlanations framework proposed by Lundberg and colleagues for interpreting predictions, later extended with TreeExplainer for tree-based models such as random forests and gradient boosted trees. Subsequent work has refined these approximations, with researchers noting that assumptions like feature independence can distort the true Shapley values, and improved methods such as a modified Kernel SHAP that estimates conditional distributions to account for feature dependence. In energy systems, the value has been used to allocate collective costs in residential microgrids managed by reinforcement learning, to compute redispatch congestion cost allocation in a way that preserves data privacy, and to determine the value of flexible devices in demand side management. Most of these applications, the authors note, target either computation methods or black-box models, leaving white-box models for distributed energy resources largely unexplored territory.

White-box models are built from explicit physical equations rather than learned from data, which makes them transparent but also modular: a system model can be assembled from distinct submodels, each describing a component or process. The research team framed the question of which submodels deserve to be included as a four-player cooperative game. The players were the mathematical submodels, and the value of any coalition of submodels was defined as the maximum adjustable power of the resulting system model multiplied by its accuracy. Accuracy was measured by comparing simulated load profiles over a seven-day, 168-hour horizon against real measurements using the normalized Mean Absolute Error, with model utility expressed as one minus this error, scaled to a percentage. The maximum power entered the definition because a device’s ability to contribute to load profile adjustment in demand side management is closely tied to its maximum power rating.

The case study centered on a heat pump system with two hot water storages, installed in a stand-alone house in Switzerland and equipped with automatic data measurement and storage equipment. Heat transfer occurs in three main parts of the system: the borehole ground heat exchanger, the heat pump itself, and the hot water storages. The borehole is modeled through the relationship between inlet and outlet temperatures, the absorbed thermal power, the specific heat capacity of the brine, and the brine mass flow. The heat pump’s thermal power follows from the first law of thermodynamics, combining the specific heat capacity of water, the water flow rate, the water density, and the difference between supply and return temperatures. The thermal energy change in each storage is likewise modeled thermodynamically, using the storage volume and temperature readings from sensors positioned at different heights within the tanks.

From these component equations, the researchers constructed four system models of varying complexity. Model A accounts for the energy changes in both storages. Model B neglects the small hot water tank, Model C neglects the large one, and Model D ignores the energy changes in both storages. The heat pump equations themselves are required in all four models. Treating the submodels as players in a game with four participants, the team computed the Shapley value of each one exactly, a feasible task because the number of computational steps remains manageable at this scale. The full calculation procedure for the first submodel, which represents the heat pump equations essential to every system model, is laid out in the paper, yielding a Shapley value of 10.58 kilowatts when rounded to two decimal places.

The remaining three submodels received Shapley values of 0.68, 0.25, and 0.52 kilowatts respectively. The efficiency property provided a direct check on the arithmetic: summing all four values gives exactly 12.03 kilowatts, matching the value of the complete coalition. The other three properties did not apply in this use case, since no two submodels had equal Shapley values, the value function was uniquely defined, and every submodel contributed something to the coalition. The dominant result was expected, the authors write, because the first submodel is essential across all system models, giving it a marked influence on the formulation of the most complete model when that model is used to evaluate the flexibility of the investigated demand side management assets.

The most striking comparison involved the two storage submodels. The large hot water storage holds 500 liters, while the small one holds 360 liters, a capacity ratio of 1.39. Yet the Shapley value of the large storage submodel came out at 2.08 times that of the small storage submodel. This means the contribution of the storage submodel is not linearly proportional to its physical capacity. The authors draw a practical implication: when selecting submodels to construct a system model while controlling model complexity, the influence of smaller storage units can be selectively neglected. In other words, a modeler aiming to trim computational cost may be able to drop the smaller tank’s submodel with less damage to utility than its physical size alone would suggest.

The authors are careful to delineate what the study does not resolve. The system models A through D take different hardware configurations into account, so the question of whether model complexity can be varied without altering the system structure remains open, and addressing this limitation is flagged as an important avenue for future work. If the overall system structure stays unchanged, whether the maximum power or another factor should enter the value function definition also warrants further investigation. The feasibility of the approach has been demonstrated for heat pump systems within a white-box modeling framework, but its transferability to other white-box energy system models and configurations remains to be investigated. The team also distinguishes the method from sensitivity analysis, which apportions uncertainty in a model’s output to uncertainty in its inputs and is better suited as a prerequisite for statistical models or a robustness test, rather than the contribution-quantification role the Shapley value plays here.

In their conclusion, the researchers position the method as a complement to a previously proposed hypothesis that the complexity-utility relationship in demand side management modeling generally follows a diminishing marginal utility curve. Used together, the two tools could help developers, engineers, and operators not only select a model with an appropriate level of complexity based on utility requirements, but also understand which submodel within a system model contributes most to overall utility, particularly when the system model can be viewed as a coalition. Future work could redefine the value function to integrate variables such as energy costs, repeat the evaluation under identical system conditions to reduce confounding influences, and extend the approach to other model types such as lumped resistance-capacity models or gray-box models. No datasets were generated or analyzed during the study, the authors report, with the demonstration resting on the modeling framework and case study described in the paper.

Subject of Research: Quantifying the contribution of submodels in white-box distributed energy resource models for demand side management using the Shapley value

Article Title: Assessing modeling choices for distributed energy resources in demand side management based on the Shapley value

Article References: Li, C., Förderer, K., Matthes, J., & Hagenmeyer, V. (2026). Assessing modeling choices for distributed energy resources in demand side management based on the Shapley value. Energy Informatics, 9(1), Article 64. https://doi.org/10.1186/s42162-026-00680-3

Image Credits: AI Generated

DOI: 10.1186/s42162-026-00680-3

Keywords: Shapley value, distributed energy resources, demand side management, white-box models, heat pump, hot water storage, model complexity, cooperative game theory, energy flexibility, model accuracy, load profile, Assessing