⚡ electrical engineering

Grid-Compatible Flexibility from Multi-Energy Systems via Cyclic-Terminal Economic MPC with Hybrid Thermal-Electrical Dynamics

This paper presents a unified Economic Model Predictive Control framework for multi-energy systems that integrates hybrid thermal-electrical dynamics and market signals, demonstrating that a sufficiently large terminal penalty weight effectively substitutes for long prediction horizons to ensure optimal closed-loop performance and computational tractability.

Original authors: Azzam Abdul, Schwenkel Lukas, Scheurer Leon, Häbig Pascal, Hufendiek Kai

Published 2026-08-10

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CC BY 4.0

Original authors: Azzam Abdul, Schwenkel Lukas, Scheurer Leon, Häbig Pascal, Hufendiek Kai

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the energy grid as a giant, bustling city where electricity and heat are the two main currencies. In the past, these currencies were handled by separate banks: one for lights and gadgets, another for warm showers and heated buildings. But as the world shifts toward green energy, these banks need to talk to each other. The problem is that electricity is like cash in your pocket—it's great for quick transactions but hard to stash for later. Heat, on the other hand, is like a heavy, warm stone; it takes time to heat up, but once it's hot, it stays warm for a long time, acting as a natural battery.

To run this city efficiently, we need a super-smart traffic controller that can predict the future. This is where Model Predictive Control (MPC) comes in. Think of it as a chess player who doesn't just look at the next move, but plans several turns ahead, constantly adjusting the strategy as the game changes. When this controller's goal is to save money rather than just follow a fixed schedule, it's called Economic MPC (EMPC). The big challenge scientists face is making this controller fast enough to run in real-time while juggling the complex, slow-moving physics of heat with the lightning-fast dynamics of electricity, all while reacting to fluctuating prices.

This paper introduces a clever new way to run this controller for a campus-sized energy system that mixes heat and power. The researchers built a unified "brain" that treats the heating pipes and electrical wires as one single, interconnected system. Instead of trying to guess the perfect future every time, they used a trick: they calculated a "perfect weekly schedule" offline (like a master plan for a typical week) and then used that plan as a guidepost for the real-time controller. They didn't force the system to stick to the plan rigidly; instead, they added a "soft tether" that gently pulls the system back toward the plan if it starts to drift too far.

The team ran thousands of simulations on a virtual campus equipped with solar panels, batteries, heat pumps, and a combined heat-and-power plant. They discovered something surprising: the controller didn't need to look very far into the future to be effective. Usually, you'd think looking further ahead (a longer "prediction horizon") would always be better. However, they found that if you have a strong enough "soft tether" (a high terminal weight) anchoring the system to the weekly plan, the length of the look-ahead window barely matters. A short look-ahead of just 20 minutes performed almost as well as a look-ahead of 72 hours, provided the tether was strong enough.

In fact, the paper argues that the "look-ahead time" and the "tether strength" act as substitutes. Without the tether, the controller needs to look ahead for two or three full days (about 48 to 72 hours) to figure out the best rhythm on its own. But with the tether, it can get the same result looking only a few minutes ahead. This is a huge win for speed, meaning the computer doesn't have to do heavy lifting to solve complex math problems every few minutes.

However, the study also uncovered a quirky behavior in the thermal storage (a giant tank of hot water). While the controller kept the battery and the heating network perfectly aligned with the weekly plan, the hot water tank seemed to "drift" over time, slowly filling up or emptying out in a way the weekly plan didn't predict. The researchers explain this isn't a mistake, but a quirk of the math: because the cost of holding hot water is essentially zero, the controller feels free to let the tank wander, as long as it doesn't break any rules. This drift happens even when the system is saving money effectively.

The researchers tested their system under perfect conditions where they knew the future weather and prices exactly. In these simulations, the online controller managed to keep the total cost within less than 1% of the theoretical best possible cost. They found that for this specific campus setup, a prediction horizon of 24 hours (288 steps) is the sweet spot: it's long enough to see a full day of price changes and keep the system stable, but short enough to solve quickly. They recommend a specific "tether weight" of 1,000, which keeps the system anchored without being too rigid.

Ultimately, the paper shows that by using a smart anchor (the periodic reference) and a flexible tether, we can make complex energy systems run efficiently without needing supercomputers to look days into the future. While the results are based on simulations and assume perfect knowledge of the future, the findings suggest a practical path forward for managing the messy, mixed world of future green energy grids.

Technical Summary: Grid-Compatible Flexibility from Multi-Energy Systems via Cyclic-Terminal Economic MPC

Problem Statement

Heating and cooling sectors remain heavily dependent on fossil fuels, posing a significant barrier to decarbonization, while offering large-scale inherent storage via thermal inertia that power grids lack. Coordinating these thermal and electrical domains introduces operational challenges due to the volatility of renewable generation, demand uncertainties, and the distinct dynamics of assets like Combined Heat and Power (CHP) plants, heat pumps (HPs), batteries, and Thermal Energy Storage (TES).

Conventional approaches, such as rule-based dispatch or static Mixed-Integer Linear Programming (MILP) scheduling, often treat assets in isolation or rely on fixed schedules. These methods lack the ability to anticipate future states or price signals and cannot close the loop to react to forecast errors. While Model Predictive Control (MPC) offers a solution by optimizing over a receding horizon, applying it to integrated multi-energy systems is computationally demanding. Furthermore, standard Economic MPC (EMPC) formulations often struggle to ensure stability and performance guarantees without imposing hard terminal constraints, which can render the optimization infeasible under time-varying disturbances.

Methodology

The paper proposes a unified Cyclic-Terminal Economic Model Predictive Control (EMPC) framework designed for the coordinated operation of integrated thermal and electrical networks.

1. Unified State-Space Representation:
The authors develop a single mixed-integer state-space model that couples:

District Heating Network (DHN): Modeled using a reduced-order Resistive–Capacitive (RC) approximation, focusing on the thermal storage effect of the network water mass while neglecting hydraulic pressure dynamics.
Electrical Grid: Modeled using a linearized DC power flow (DCPF) approximation.
Assets: Includes a stratified TES, a battery, a large-scale HP, and a CHP plant. The CHP is formulated using a Mixed Logical Dynamical (MLD) approach to handle its hybrid on/off logic and startup constraints directly within the optimization.
Disturbances: Time-varying signals (solar irradiance, ambient temperature, demand, electricity prices) are treated as exogenous inputs.

2. Cyclic-Terminal Formulation:
Instead of tracking a fixed setpoint, the EMPC minimizes an economic stage cost (fuel, electricity, degradation, and curtailment costs). To ensure stability and performance without hard terminal sets, the framework utilizes a periodic economic reference orbit:

Offline: An optimal periodic state-input trajectory is computed once offline over a nominal period (e.g., 24 hours) assuming periodic disturbances.
Online: The receding-horizon controller is anchored to this fixed reference using soft terminal constraints. A scaled slack penalty is applied to deviations between the terminal state and the reference orbit. This allows the controller to maintain feasibility when realized disturbances deviate from the nominal estimate while still guiding the system toward the economically optimal periodic regime.

3. Experimental Setup:
The framework is tested on a campus-scale multi-energy system (University of Stuttgart) under perfect foresight conditions. The simulation spans three days with a 300-second sampling time and a 24-hour prediction horizon. A joint sweep is performed across 14 prediction horizon lengths ( $N$ ) and 8 terminal penalty weights ( $w_{term}$ ) to analyze the interaction between these design parameters.

Key Contributions

Unified Hybrid Model: A single affine state-space model coupling an RC-based DHN and a DC power flow grid with five generation/storage components (CHP, HP, TES, Battery, Solar) in a mixed-integer linear program with a convex economic cost.
Cyclic-Terminal EMPC Design: A practical implementation using scaled soft terminal constraints that anchor the receding-horizon operation to an offline periodic economic orbit, exploiting daily price and demand periodicity without requiring hard terminal sets.
Sensitivity Analysis of Design Parameters: A comprehensive closed-loop sensitivity analysis characterizing the relationship between the prediction horizon and the terminal penalty weight. The study identifies that these two parameters act as substitutes rather than independent tuning knobs.
Empirical Characterization of Storage Drift: The identification and interpretation of a "receding-horizon drift" in the thermal energy storage state, where the online controller drives the storage far across its range while the offline periodic reference keeps it nearly inert.

Results

1. Orbit Alignment and Cost Performance:

Substitution Effect: The study demonstrates that a sufficiently large terminal weight ( $w_{term} \ge 300$ ) allows the system to achieve near-optimal performance (within 0.4% to 1.0% of the offline reference cost) even at the shortest prediction horizons (20 minutes). Without this anchoring ( $w_{term}=0$ ), the prediction horizon must span two to three diurnal cycles (48–72 hours) to achieve similar alignment.
Cost Plateau: Once the terminal weight exceeds an activation threshold, the closed-loop cost becomes insensitive to the weight over a wide range (a "multi-decade plateau"). Similarly, with the terminal condition active, the prediction horizon ceases to be a performance-critical parameter and becomes purely a computational one.
Baseline Performance: With a 24-hour horizon and a moderate terminal weight ( $w_{term}=10^3$ ), the online EMPC tracks the offline periodic orbit closely, with a cumulative cost gap of only 0.77% over the simulation window.

2. Flexibility Allocation:

Asymmetric Roles: The system exhibits an asymmetric flexibility allocation. The battery acts as the medium-term mover, performing slow, multi-hour price arbitrage. Conversely, the TES operates as a fast, near-continuous cycler, primarily absorbing a persistent overproduction bias from the heat pump rather than responding directly to price signals.
CHP Dispatch: The CHP is dispatched predominantly during high-price periods, with its commitment schedule remaining largely invariant across different horizon lengths once the terminal condition is active.

3. Receding-Horizon Drift:

The TES state exhibits a persistent drift where the online controller accumulates inventory relative to the offline reference. This occurs because the TES energy level carries no direct stage cost (only passive losses), making strict dissipativity fail in that coordinate. The offline reference is forced to be inert by its hard periodicity constraint, whereas the online controller exploits the "free" storage capacity to satisfy terminal constraints via scheduled discharges that are never executed due to the shifting horizon.

Significance and Claims

The paper claims that the proposed cyclic-terminal EMPC framework successfully bridges the gap between computational tractability and economic optimality for multi-energy systems. By using a soft terminal constraint anchored to a periodic reference, the method:

Decouples Performance from Horizon Length: It allows for short prediction horizons (reducing computational burden) without sacrificing economic performance, provided a sufficiently large terminal weight is used.
Ensures Feasibility: The soft constraint mechanism preserves feasibility under disturbances where hard terminal constraints might fail.
Reveals Structural Properties: The analysis highlights that in hybrid systems with flat cost landscapes (like lossy thermal storage), receding-horizon controllers may exhibit drift even when average performance bounds are met, a phenomenon distinct from integer-switching noise.

The authors note that these findings are specific to the studied campus system and price weeks. The study assumes perfect foresight and a plant-model match, acknowledging that future work must address forecast uncertainty and model mismatch through robust or stochastic extensions. The paper does not claim universal applicability to all grid configurations but establishes a validated methodology for the specific class of integrated thermal-electrical systems modeled.