Holistic MIP Model

• Holistic MIP model is a mathematically precise framework that integrates network topology, variable definitions, and operational logic for multi-energy systems.
• It incorporates full constraint systems and piecewise-linear conversions to optimize economic, environmental, and reliability objectives within a single formulation.
• Comparative analysis of Model A and Model B reveals trade-offs between user-friendly interpretability and computational scalability for real-world applications.

A holistic MIP model denotes a mathematically precise framework for modeling integrated multi-energy systems, where the full spectrum of system structure, technical requirements, operational logic, and multi-objective performance metrics are embedded within a single mixed-integer programming (MIP) formulation. Unlike narrowly focused formulations emphasizing only mathematical constraints or parameter settings, holistic modeling interrogates critical post-formulation choices such as topology representation, variable definition, constraint assembly, objective integration, and model compactness. This approach is characterized by explicit enumeration of all modeling degrees of freedom and rigorous analysis of their impact on problem tractability, solver performance, and practical interpretability (Riedmüller et al., 20 May 2025).

1. Network Abstraction and Topology Construction

Constructing a holistic MIP model involves reducing the physical multi-energy network into an abstract directed graph $G=(V,A)$, where node sets $V$ encode generating units ($I$), storage ($K$), balance ($B$), demand ($D$), and market ($M$) nodes. Resource types ($R$) are indexed on arcs $A\subseteq V\times V$, with each arc $(v,w)$ carrying resource $r\in R$ from node $v$ to node $w$. Two distinct topological representations are admissible:

• Hierarchical Port-based (Model A): Permits auxiliary "container" nodes $C$, each encapsulating a sub-graph $G_c=(V_c,A_c)$ with dedicated inbound/outbound ports.
• Flat Arc-based (Model B): Contracts all auxiliary nodes, yielding a physical-only topology devoid of containers.

Node incidence is described by successor sets $\delta^{+}(v)$ and predecessor sets $\delta^{-}(v)$, enabling generalizable flow logic across both formulations.

2. Variable, Parameter, and Constraint Specification

Holistic modeling incorporates full account of time indices, units, resources, and operational parameters. Notation follows:

• $u_{i,t}\in\{0,1\}$: unit commitment
• $s_{i,t}\in\{0,1\}$: startup indicator
• $x_{(v,w),t}\geq 0$, $x^{in}_{v,r,t}$, $x^{out}_{v,r,t}$: flow variables (arc-based or port-based)
• $h_{k,r,t}$: storage state
• $p^r_{t,v}$, $e^r_{t,v}$: market transactions

Constraints ensure:

• Flow conservation: $$\sum_{w\in\delta^{-}(v)}x_{(w,v),t}+p^r_{t,v}=\sum_{w\in\delta^{+}(v)}x_{(v,w),t}+d^r_{t,v}+e^r_{t,v}$$
• Conversion/production: Piecewise-linear mapping $y_{i,r_2,t} = \varphi_{i,r_1\to r_2}(y_{i,r_1,t})$
• Unit regime (start-up, min up/down, ramp): All constraints are linear or MIP-linearizable.
• Storage dynamics: $$h_{k,r,t} = \text{losses}_k \cdot h_{k,r,t-1} + \eta^{in}_{k,r}\cdot \sum_{(v,k)}x_{(v,k),t} - \frac{1}{\eta^{out}_{k,r}}\cdot \sum_{(k,w)}x_{(k,w),t}$$

All technical requirements are gathered in a unified constraint system, allowing systematic analysis and improvement.

3. Multi-Objective Integration and Scalarization

Holistic MIP models support simultaneous optimization of economic, environmental, and reliability targets. Canonical objectives are:

• $f_1$: total operational cost
• $f_2$: total CO$_2$ emissions
• $f_3$: reliable heat from CHP units

Objective integration strategies include:

• Weighted sum: $\min\;\theta_1 f_1 + \theta_2 f_2 + \theta_3 f_3$
• Pareto methods: $\varepsilon$-constraint or boundary intersection
• Lexicographic: Sequential solution with objectives fixed at bounds

This explicit structuring renders trade-off surfaces and reliability criteria computationally accessible within the same topological MIP instance.

4. Topological Modeling Choices: Model A vs. Model B

Two mathematically equivalent but operationally distinct MIP constructions are possible:

• Model A: Hierarchical container nodes with explicit ports; user-friendly, visually oriented, and highly interpretable for technology-centric modeling. Redundant variable and constraint structures arise due to the port coupling, mostly removable by solver presolve.
• Model B: All auxiliary/topological nodes contracted, arc-based flow variables only; compact, minimal, and optimal for large-scale or algorithm-centric applications.

Theoretical equivalence is demonstrated via node and edge contraction (Def. 1, Lemma 1), and conversion merging (Lemma 2). Every feasible solution in Model A maps one-to-one to Model B with preserved cost and feasibility.

Table: Computational Comparison, Berlin Heating Network

Metric	Model A	Model B	Difference
Graph nodes	128	33	–74%
Graph arcs	171	38	–78%
MIP Variables	74,029	16,740	–77%
MIP Constraints	70,309	17,102	–76%
Presolve Vars	3,517	4,015	+14% (rebalancing)
Presolve Cons	4,206	4,001	–5%
Total time [s]	0.82	0.66	–20%
Presolve only [s]	0.23	0.17	–28%

Model B achieves pronounced reductions in graph size and runtime (especially with presolve off), while Model A facilitates easier end-user adjustment and interpretation (Riedmüller et al., 20 May 2025).

5. Computational and Algorithmic Implications

The choice of modeling approach directly affects solver behavior:

• Problem size: Model A introduces $\approx 2|A|$ extra variables and $|A|$ constraints (ports); Model B is strictly minimal, advantageous for branch-and-bound, decomposition, and cut-based methods.
• Presolve effectiveness: Commercial solvers (e.g. Gurobi 11) eliminate Model A's redundancy at the cost of longer presolve time but achieve comparable final solve times.
• Interpretability vs. scalability: Model A is legible for domain experts; Model B is optimal for mathematical and high-throughput optimization.

Solver features such as aggressive presolve or specialized decomposition algorithms (e.g. Benders decomposition) may prefer Model B's arc-centric structure.

6. Case Study: Berlin District Heating Network

A practical illustration utilized a realistic subarea of Berlin's district heating network (four generation units, one thermal storage, $T=180$ periods):

• Using Model B, node count declined by 74%, arcs by 78%, variables from $\sim$74k to $\sim$17k.
• Gurobi presolve reduced both models to $\sim$4k variables/4.3k constraints per lex-solve.
• Model B's runtime was on average 20% lower, with presolve-only time 28% lower; with presolve disabled, Model B ran 84% faster overall.

These results validate the trade-offs between interpretability and computational efficiency central to holistic modeling. A plausible implication is that for large, data-driven multi-energy networks, graph contraction and conversion merging should be standard, but technology-oriented scenarios may warrant explicit port-nodes for user engagement.

7. Guidelines and Best Practices

For holistic MIP modeling in integrated multi-energy systems:

• Model A should be adopted for technology-specific, visually interpretable frameworks or where users directly adjust model components.
• Model B is preferred for large-scale automation, minimal memory footprint, and advanced decomposition or cutting-plane algorithms.
• Solver presolve can mitigate the overhead of redundant constructs but at measurable cost in initialization time.
• Multi-objective optimization benefits from Model B's flat structure for Pareto surface computation.
• Piecewise-linear conversion merging should always be performed to reduce unnecessary variables and constraints.

Overall, holistic MIP modeling combines diverse technical requirements, operational logic, detailed network topologies, and rigorous multi-objective structuring within a single optimization framework, balancing human-centric clarity and machine-centric efficiency (Riedmüller et al., 20 May 2025).