Global Optimization of Flexible District Heating Networks

Abstract: District heating networks are a central tool to achieve low-carbon heat supplies. In this realm, they face the challenge of dealing with increasingly heterogeneous, partially time-varying renewable sources, thermal storage, and meshed topologies. This paper examines global optimization of the operation of such district heating networks over multiple time steps, based on a stationary, yet realistical nonlinear network model. To accelerate the solution performance of a spatial branch-and-bound algorithm for one time step, the following new methodological ingredients are introduced: exclusion of cyclic flow, symmetry exploitation between supply and return networks, reduction of temperature mixing constraints, novel primal heuristics and branching rules. The proposed methods are evaluated on a set of generated and real-world benchmark network instances with cycles and several suppliers. On the generated benchmark instances, using these methods more than doubles the number of solved instances and more than halves the runtime. For the real-world benchmark instances, the resulting algorithm produces solutions with guaranteed quality in reasonable run time. For multiple time steps that are coupled by a storage, a time decomposition approach is investigated. Under assumptions that are reasonable in practice, this approach is shown to yield an optimal solution. On a small example network, this decomposition is able to compute optimal solutions in less than a second, while solving the complete time-coupled problem is not possible within one hour.

• The paper develops a global spatial branch-and-bound framework that uses network structure, heuristics, and specialized branching to solve 50 benchmark instances versus 20 for the baseline, while reducing shifted geometric mean runtime from 2,342 to about 959 seconds.
• The paper proves that internal supply and return flows are acyclic and symmetric under mirrored network assumptions, enabling valid cycle-exclusion constraints and elimination of roughly half the pipe-flow variables, with an average reduction of 72.1 arcs per instance.
• The paper introduces a storage-scheduling decomposition that reaches the optimum under explicit assumptions and solves a four-period test in under 0.4 seconds, although larger decentralized networks and less restrictive storage configurations remain challenging.

This paper develops a global optimization framework for the operational scheduling of district heating networks (DHNs), targeting the non-convex mixed-integer nonlinear programs (MINLPs) that arise from a stationary but physically realistic network model. The work addresses two problem classes: single time-step operation of meshed networks with multiple suppliers, and multi-period scheduling coupled through thermal storage. The authors contribute structural theory (acyclicity of flows, supply/return symmetry), algorithmic enhancements to a spatial branch-and-bound solver built on SCIP, and a provably optimal decomposition scheme for the storage-coupled case under explicit assumptions.

Stationary MINLP model

The DHN is modeled as a directed graph split into a supply and a return part, connected by heating arcs (suppliers) and demand arcs (consumers). Decision variables include mass flow rates, nodal pressures and temperatures, arc start/end temperatures, binary flow-direction variables, heat power, and pump power. Hydraulics follow the stationary momentum equation with a Nikuradse friction factor; thermal behavior uses a first-order Taylor approximation of the exponential temperature-decay equation along pipes, which the authors show is accurate to below 1 °C for mass flows above 1.5 kg/s. Temperature mixing at nodes is expressed as nonlinear equality constraints whose form depends on flow directions, and demands are enforced via fixed return temperature together with a minimum pressure drop across consumers. The objective minimizes the sum of electric pump power and heat energy at suppliers.

The resulting program is non-convex for two reasons highlighted by the authors: heat demand can be met by any product combination of temperature and flow rate, and temperature mixing constraints depend on binary flow directions. A degree-of-freedom count shows the model has $2|A^H|$ more variables than independent equations, reflecting genuine operational freedom at suppliers.

Structural properties

Two theoretical results drive the algorithmic design. First, a lemma establishes that no nonzero flow can circulate along a cycle contained entirely in the supply or return part: summing pressure-drop equations around such a cycle forces the sum of $\zeta_a q_a |q_a|$ terms to vanish, which is impossible for a consistent circulation. This yields valid inequalities on the binary direction variables that exclude cyclic flow patterns.

Second, under the assumption that supply and return topologies are mirror images with identical pipe parameters, the paper proves that corresponding supply and return arcs carry identical mass flow. The proof proceeds by an exchange argument: any discrepancy propagates along paths until either a leaf node contradicts flow conservation or a cycle contradicts the acyclicity lemma. These equalities allow roughly half of the pipe variables to be eliminated — on average 72.1 arcs per instance in the experiments.

Algorithmic enhancements and single-period results

The implementation extends SCIP's spatial branch-and-bound (LP relaxations via McCormick and gradient cuts, spatial branching) with five ingredients: cycle-exclusion constraints generated via depth-first search over fundamental cycles; exploitation of supply/return flow equality; simplification of temperature-mixing constraints after presolving fixes flow directions (on average 86.6 constraints simplified per instance); two primal heuristics (one fixing binaries from LP solutions and solving the residual NLP, one solving a hydraulics-only subproblem first); and branching rules that fix the demand arc attaining the minimal supply–return pressure difference, exploiting the fact that positive pump costs force this difference to be tight somewhere.

On 26 generated instances (10–150 supply nodes, up to 30 suppliers, up to 4 cycles, four load scenarios each), the combined configuration solves 50 instances versus 20 for the baseline within a two-hour limit, and reduces shifted geometric mean runtime from 2342 s to about 959 s — i.e., more than doubling solved instances while cutting runtime by more than half. Individually, cycle exclusion has the largest effect (29 solved), followed by the hydraulic heuristic. Among branching variants, BranchTree matches Combined with a further 3% speedup, while BranchAll solves fewer instances; the choice depends on how many networks are tree-shaped.

On four real-world-inspired networks derived from Darmstadt data (contracted pipes, aggregated district demands; "today" centralized and "future" decentralized scenarios with 45–85 nodes), both tested variants close DaNo2020 and TU2020 to within the 1% gap limit (294–390 s). For the harder future scenarios DaNo2030 and TU2030, neither variant proves optimality in five hours; BranchAll achieves substantially smaller gaps (8.07% vs. 29.98% for DaNo2030) but is slower on the solved instances. Notably, TU2030 exhibits an enormous gap (3465% for Combined), indicating that the dual bound converges very slowly on some decentralized configurations — a limitation the authors do not resolve.

Storage-coupled scheduling and decomposition

For $T$ time steps coupled only through storage fill levels, the full time-coupled MINLP is computationally prohibitive. The authors propose solving first the storage-free problem, which decomposes into $T$ independent single-period problems, then determining an optimal charging/discharging schedule in a second small model that redistributes energy between each storage and its parallel heating arc.

Optimality of this scheme is proven under Assumption 1: uniform pump efficiency; each storage parallel to exactly one heating arc with unbounded heat supply; unique connecting arcs with positive flow direction (so storages charge only from their heating, not from the network, though they may discharge into it); and equal electricity costs for storage and heating pumps. The proof constructs, from any feasible solution of the full problem, feasible solutions of both decomposition stages with no worse objective, using temperature-mixing identities at the storage/heating junctions. The assumptions are acknowledged as restrictive — notably the prohibition on charging from the network — though the authors argue they cover practically relevant configurations such as CHP-coupled central storages, and note the method remains usable as a heuristic otherwise.

On a minimal four-period example with time-varying costs and demands, the decomposition finds the optimum in under 0.4 s total, whereas the monolithic problem hits a one-hour time limit without a usable dual bound; the decomposed objective (403.718) matches the monolithic best solution (403.637) within accuracy. This demonstrates that the decomposition enables solvability where the direct approach fails entirely, although the evidence rests on a single, deliberately simple network.

Limitations and open questions

Several limitations are stated explicitly. The Taylor approximation of pipe temperature decay degrades for small mass flows, and the induced lower bound on flow magnitude constrains the feasible set. Cycle exclusion covers only fundamental cycles found by DFS, which suffices for most real-world topologies but not all. Real-world results show that instances with many cycles and suppliers (the 2030 scenarios) remain far from proven optimality, with one instance exhibiting a gap above 3400%. The decomposition's optimality guarantee requires the parallel-storage assumptions; whether it holds under network charging, restricted supplier capacity, or heterogeneous pump efficiencies is open. The evaluation of the decomposition as a general heuristic beyond the toy example is left unaddressed, as is scaling to larger networks and longer horizons.

Conclusion

The paper demonstrates that globally optimal operation of meshed, multi-supplier DHNs is attainable in practice when structural properties — acyclicity and supply/return symmetry — are exploited inside spatial branch-and-bound, yielding certified solutions rather than heuristic ones. For storage-coupled scheduling, a decomposition with proven optimality under stated assumptions reduces solve times from intractable to sub-second on the tested example. The remaining challenges identified by the authors are larger instances, storage settings violating the decomposition assumptions, and systematic evaluation of the decomposition as a heuristic.