# Handling Non-Linearities in Modelling the Optimal Design and Operation of a Multi-Energy System

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions [1,3]. Accordingly, the optimization of such systems has become a crucial research topic and should help answer questions related to the efficient integration of renewable energy sources (RES) or increased uses of energy storage solutions.

_{2}emission rate [11,12,14,15], or a minimization of the environmental cost [13]. The optimization of the costs can be calculated as the minimization of total costs [14,16], the minimization of the cost of power generation [13], the maximization of profits [17,18], or the minimization of costs compared to a reference system [7,19].

## 2. Problem Definition and Formulation

- a combined heat and power unit (CHP) which provides heat and electricity from gas;
- an electric boiler (EB) which provides heat;
- a gas boiler (GB), which also provides heat;
- photovoltaic (PV) panels which generate electricity;
- solar thermal (ST) panels which generate heat.

## 3. Solution Approach

#### 3.1. Constant Efficiency

#### 3.2. Classical Piecewise Linearization Method

- ${{\mathit{\alpha}}_{\mathit{m},\mathit{n},\mathit{t}}}\in [0,1]$ which represent the coefficients of the convex combination $(\forall m=1,\dots ,M,n=1,\dots ,N,t\in \mathcal{T})$
- ${{\mathit{h}}_{\mathit{m},\mathit{n},\mathit{t}}^{\mathit{u}}}\in \{0,1\}$ and ${{\mathit{h}}_{\mathit{m},\mathit{n},\mathit{t}}^{\mathit{l}}}\in \{0,1\}$ that indicate which triangle is selected $(\forall m=1,\dots ,M-1,n=1,\dots ,N-1,t\in \mathcal{T})$

#### 3.3. Proposed Adapted Piecewise Linearization Method

- ${{\mathit{\alpha}}_{\mathit{n},\mathit{t}}}\in [0,1]$ ($\forall n\in 1,\dots ,N,t\in \mathcal{T}$)
- ${{\mathit{h}}_{\mathit{n},\mathit{t}}}\in \{0,1\}$ ($\forall n\in 1,\dots ,N-1,t\in \mathcal{T}$)

#### 3.4. Multi-Objective Resolution

#### 3.5. Discussion on the CHP Approximations

## 4. Case Study

## 5. Experimentation

#### 5.1. Experimental Design

^{5}s), and for the adapted method, the number of triangles varies from 1, …, 36 (when possible in less than 10

^{5}s ). For each resolution, a set of Pareto-optimal solutions $\mathcal{S}$ is obtained using the $\u03f5$-constraints method. Furthermore, we use three different performance indicators for these tests: the computation time, the mean distance of the Pareto front to the origin to evaluate the evolution of the Pareto fronts with the number of triangles, and a measure of the approximation error. The computation time for one resolution is given by the sum of the computation times of all the $\left|\mathcal{S}\right|$ steps of the $\u03f5$-constraints method.

#### 5.2. Results

#### 5.2.1. Numerical Analysis

^{6}s (i.e., 26 days) over the full year with 15 triangles, whereas it required at most 3.3 × 10

^{3}s (i.e., 55 min) over one week for the same number of triangles. In addition, it was not possible to solve the system with more than one triangle using the literature method.

#### 5.2.2. Energy Analysis

## 6. Conclusions and Perspectives

#### 6.1. Summary of the Contributions

#### 6.2. Discussion

^{4}s, the adapted method’s error is 80% lower than the method from the literature. The analysis for a full year confirms these results. Even by allowing an optimality gap when obtaining the solutions, it has not been possible to solve the model with the method from the literature for more than a single triangle per variable, while the adapted method solved the model with up to 15 triangles. Comparing the energy over the year of the proposed linearized method and the simplified constant method, we can see a relative difference of 9% in the output power of the CHP, and up to 10.6% relative difference in the values of the objective functions, showing that a more accurate approximation yields more realistic solutions to the optimization problem.

#### 6.3. Limitations and Perspectives

^{6}s to optimize the MES with an optimality gap of 0.1%. A possible improvement to the proposed linearization method would be to choose more carefully the location of the breakpoints used for linearization, so as to have more breakpoints where the function is less linear, and fewer breakpoints where it is more linear.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Shabanpour-Haghighi, A.; Karimaghaei, M. An overview on multi-carrier energy networks: From a concept to future trends and challenges. Int. J. Hydrog. Energy
**2022**, 47, 6164–6186. [Google Scholar] [CrossRef] - Beigvand, S.D.; Abdi, H.; Scala, M.L. Multicarrier energy systems. In Handbook of Energy Economics and Policy; Academic Press: Cambridge, MA, USA, 2021; pp. 433–519. [Google Scholar] [CrossRef]
- Alabi, T.M.; Agbajor, F.D.; Yang, Z.; Lu, L.; Ogungbile, A.J. Strategic potential of multi-energy system towards carbon neutrality: A forward-looking overview. Energy Built Environ.
**2023**, 4, 689–708. [Google Scholar] [CrossRef] - Ghanbari, A.; Karimi, H.; Jadid, S. Optimal planning and operation of multi-carrier networked microgrids considering multi-energy hubs in distribution networks. Energy
**2020**, 204, 117936. [Google Scholar] [CrossRef] - Fonseca, J.D.; Commenge, J.M.; Camargo, M.; Falk, L.; Gil, I.D. Multi-criteria optimization for the design and operation of distributed energy systems considering sustainability dimensions. Energy
**2021**, 214, 118989. [Google Scholar] [CrossRef] - Bischi, A.; Taccari, L.; Martelli, E.; Amaldi, E.; Manzolini, G.; Silva, P.; Campanari, S.; Macchi, E. A detailed MILP optimization model for combined cooling, heat and power system operation planning. Energy
**2014**, 74, 12–26. [Google Scholar] [CrossRef] - Adihou, Y.; Mabrouk, M.T.; Haurant, P.; Lacarrière, B. A multi-objective optimization model for the operation of decentralized multi-energy systems. J. Phys. Conf. Ser.
**2019**, 1343, 012104. [Google Scholar] [CrossRef] - Geidl, M.; Andersson, G. Optimal Power Flow of Multiple Energy Carriers. IEEE Trans. Power Syst.
**2007**, 22, 145–155. [Google Scholar] [CrossRef] - Li, K.; Ding, Y.Z.; Ai, C.; Sun, H.; Xu, Y.P.; Nedaei, N. Multi-objective optimization and multi-aspect analysis of an innovative geothermal-based multi-generation energy system for power, cooling, hydrogen, and freshwater production. Energy
**2022**, 245, 123198. [Google Scholar] [CrossRef] - Wu, M.; Du, P.; Jiang, M.; Goh, H.H.; Zhu, H.; Zhang, D.; Wu, T. An integrated energy system optimization strategy based on particle swarm optimization algorithm. Energy Rep.
**2022**, 8, 679–691. [Google Scholar] [CrossRef] - Gabrielli, P.; Gazzani, M.; Martelli, E.; Mazzotti, M. Optimal design of multi-energy systems with seasonal storage. Appl. Energy
**2018**, 219, 408–424. [Google Scholar] [CrossRef] - Weber, C.I. Multi-Objective Design and Optimization of District Energy Systems Including Polygeneration Energy Conversion Technologies. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2008. [Google Scholar] [CrossRef]
- Shan, J.; Lu, R. Multi-objective economic optimization scheduling of CCHP micro-grid based on improved bee colony algorithm considering the selection of hybrid energy storage system. Energy Rep.
**2021**, 7, 326–341. [Google Scholar] [CrossRef] - Zhao, Y.; Zhou, M.; Yue, Z.; Tan, T.; Zheng, M. Prospective optimization of CCHP system under multi-scenarios. Energy Rep.
**2022**, 8, 952–958. [Google Scholar] [CrossRef] - Fazlollahi, S.; Mandel, P.; Becker, G.; Maréchal, F. Methods for multi-objective investment and operating optimization of complex energy systems. Energy
**2012**, 45, 12–22. [Google Scholar] [CrossRef] - Wirtz, M.; Hahn, M.; Schreiber, T.; Müller, D. Design optimization of multi-energy systems using mixed-integer linear programming: Which model complexity and level of detail is sufficient? Energy Convers. Manag.
**2021**, 240, 114249. [Google Scholar] [CrossRef] - Mancarella, P. MES (multi-energy systems): An overview of concepts and evaluation models. Energy
**2014**, 65, 1–17. [Google Scholar] [CrossRef] - Piacentino, A.; Cardona, F. EABOT—Energetic analysis as a basis for robust optimization of trigeneration systems by linear programming. Energy Convers. Manag.
**2008**, 49, 3006–3016. [Google Scholar] [CrossRef] - Yousefi, H.; Ghodusinejad, M.H.; Noorollahi, Y. GA/AHP-based optimal design of a hybrid CCHP system considering economy, energy and emission. Energy Build.
**2017**, 138, 309–317. [Google Scholar] [CrossRef] - Poncelet, K.; Delarue, E.; Six, D.; Duerinck, J.; D’haeseleer, W. Impact of the level of temporal and operational detail in energy-system planning models. Appl. Energy
**2016**, 162, 631–643. [Google Scholar] [CrossRef] - Salpakari, J.; Mikkola, J.; Lund, P.D. Improved flexibility with large-scale variable renewable power in cities through optimal demand side management and power-to-heat conversion. Energy Convers. Manag.
**2016**, 126, 649–661. [Google Scholar] [CrossRef] - Milan, C.; Stadler, M.; Cardoso, G.; Mashayekh, S. Modeling of non-linear CHP efficiency curves in distributed energy systems. Appl. Energy
**2015**, 148, 334–347. [Google Scholar] [CrossRef] - Teichgraeber, H.; Brandt, A.R. Time-series aggregation for the optimization of energy systems: Goals, challenges, approaches, and opportunities. Renew. Sustain. Energy Rev.
**2022**, 157, 111984. [Google Scholar] [CrossRef] - Puchinger, J.; Raidl, G.R. Combining Metaheuristics and Exact Algorithms in Combinatorial Optimization: A Survey and Classification. In Artificial Intelligence and Knowledge Engineering Applications: A Bioinspired Approach; Springer: Berlin/Heidelberg, Germany, 2005; pp. 41–53. [Google Scholar] [CrossRef]
- Fischetti, M.; Fischetti, M. Matheuristics. In Handbook of Heuristics; Martí, R., Pardalos, P.M., Resende, M.G.C., Eds.; Springer: Cham, Switzerland, 2018; pp. 121–153. [Google Scholar] [CrossRef]
- Tesio, U.; Guelpa, E.; Verda, V. Including thermal network operation in the optimization of a Multi Energy System. Energy Convers. Manag.
**2023**, 277, 116682. [Google Scholar] [CrossRef] - Raidl, G.R.; Puchinger, J. Combining (Integer) Linear Programming Techniques and Metaheuristics for Combinatorial Optimization. In Hybrid Metaheuristics; Springer: Berlin/Heidelberg, Germany, 2008; pp. 31–62. [Google Scholar] [CrossRef]
- D’Ambrosio, C.; Lodi, A.; Martello, S. Piecewise linear approximation of functions of two variables in MILP models. Oper. Res. Lett.
**2010**, 38, 39–46. [Google Scholar] [CrossRef] - Kotzur, L.; Nolting, L.; Hoffmann, M.; Groß, T.; Smolenko, A.; Priesmann, J.; Büsing, H.; Beer, R.; Kullmann, F.; Singh, B.; et al. A modeler’s guide to handle complexity in energy systems optimization. Adv. Appl. Energy
**2021**, 4, 100063. [Google Scholar] [CrossRef] - Yokoyama, R.; Shinano, Y.; Taniguchi, S.; Ohkura, M.; Wakui, T. Optimization of energy supply systems by MILP branch and bound method in consideration of hierarchical relationship between design and operation. Energy Convers. Manag.
**2015**, 92, 92–104. [Google Scholar] [CrossRef] - Klemm, C.; Vennemann, P. Modeling and optimization of multi-energy systems in mixed-use districts: A review of existing methods and approaches. Renew. Sustain. Energy Rev.
**2021**, 135, 110206. [Google Scholar] [CrossRef] - Haimes, Y.Y. Integrated System Identification and Optimization. In Control and Dynamic Systems; Academic Press: Cambridge, MA, USA, 1973; Volume 10, pp. 435–518. [Google Scholar] [CrossRef]
- Gudmundsson, O.; Thorsen, J.E.; Zhang, L. Cost analysis of district heating compared to its competing technologies. In Proceedings of the WIT Transactions on Ecology and the Environment, Bucharest, Romania, 19–21 June 2013; WIT Press: Southampton, UK, 2013; Volume 176, pp. 3–13. [Google Scholar] [CrossRef]
- Armila, N. Electrification as an Alternative for Combustion Technologies in Existing Finnish District Heating Networks. Master’s Thesis, Aalto University, School of Electrical Engineering, Espoo, Finland, 2020. [Google Scholar]
- Ashfaq, A.; Ianakiev, A. Cost-minimised design of a highly renewable heating network for fossil-free future. Energy
**2018**, 152, 613–626. [Google Scholar] [CrossRef]

**Figure 7.**Adapted piece-wise linearization for two different installed capacities. (

**a**) Installed capacity ${P}_{CHP,nom}=200\mathrm{k}{\mathrm{W}}_{\mathrm{e}}$. (

**b**) Installed capacity ${P}_{CHP,nom}=1000\mathrm{k}{\mathrm{W}}_{\mathrm{e}}$.

**Figure 8.**Time series parameters: (

**a**) energy demands (${L}_{e,t}$, ${L}_{h,t}$) and (

**b**) environmental parameters (${T}_{a,t}$, ${G}_{\beta ,t}$).

**Figure 10.**Parts of the function to be approximated for the three weeks: (

**a**) summer, (

**b**) mid-season and (

**c**) winter.

**Figure 12.**Objective functions (average and standard deviation) for the (

**a**) summer week, (

**b**) mid-season week (

**c**) and winter week.

**Figure 14.**Indicators on the full year (gap 0.1%): (

**a**) mean cumulative errors, (

**b**) mean distances, (

**c**) computation times and (

**d**) objective functions.

**Figure 15.**Efficiency ${\eta}_{e,CHP}$ for 10 solutions using the adapted method with 9 triangles, for the (

**a**) summer week, (

**b**) mid-season week and (

**c**) winter week.

**Figure 18.**Outputs for two days of the year for the adapted linearization method with 15 triangles: (

**a**) solution (1), (

**b**) solution (6), (

**c**) solution (10).

Notation | Description |
---|---|

$\mathcal{N}=\{e,h,g\}$ | Set of energy carriers, e (electricity), h (heat), g (gas, only used as fuel) |

$\mathcal{M}=\{CHP,GB,EB,PV,ST\}$ | Set of technologies |

$\mathcal{T}=\{1,2,\dots ,8760\}$ | Set of hours of the year |

${P}_{i,j,t}$ | Energy of type $i\in \mathcal{N}$ generated by technology $j\in \mathcal{M}$ at time $t\in \mathcal{T}$ |

${F}_{i,j,t}$ | Energy of type $i\in \mathcal{N}$ absorbed by technology $j\in \mathcal{M}$ (as fuel) at time $t\in \mathcal{T}$ |

${U}_{i,t}$ | Energy of type $i\in \mathcal{N}$ imported at time $t\in \mathcal{T}$ |

${V}_{i,t}$ | Energy of type $i\in \mathcal{N}$ exported at time $t\in \mathcal{T}$ |

${L}_{i,t}$ | Demand of energy of type $i\in \mathcal{N}$ at time $t\in \mathcal{T}$ |

Variable | Description |
---|---|

${{\mathit{P}}_{\mathit{CHP},\mathit{nom}}}$ | Electrical rated power of the CHP ($\mathrm{k}{\mathrm{W}}_{\mathrm{e}}$, [${P}_{CHP,nom}^{min},{P}_{CHP,nom}^{max}$]) |

${{\mathit{P}}_{\mathit{GB},\mathit{nom}}}$ | Thermal rated power of the GB ($\mathrm{k}{\mathrm{W}}_{\mathrm{th}}$, [${P}_{GB,nom}^{min},{P}_{GB,nom}^{max}$]) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{EB},\mathit{nom}}}\end{array}$ | Thermal rated power of the EB ($\mathrm{k}{\mathrm{W}}_{\mathrm{th}}$, [${P}_{EB,nom}^{min},{P}_{EB,nom}^{max}$]) |

$\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{PV}}}\end{array}$ | Area of PV (${\mathrm{m}}^{2}$, [${A}_{PV}^{min},{A}_{PV}^{max}$]) |

$\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{ST}}}\end{array}$ | Area of ST (${\mathrm{m}}^{2}$, [${A}_{ST}^{min},{A}_{ST}^{max}$]) |

Variable | Description |
---|---|

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{CHP},\mathit{t}}}\end{array}$ | Electricity generated by the CHP, $\forall t\in \mathcal{T}$ |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{CHP},\mathit{t}}}\end{array}$ | Heat generated and used by CHP (extra heat is lost), $\forall t\in \mathcal{T}$ |

$\begin{array}{c}\hfill {{\mathit{F}}_{\mathit{g},\mathit{CHP},\mathit{t}}}\end{array}$ | Gas used as fuel by the CHP, $\forall t\in \mathcal{T}$ |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{GB},\mathit{t}}}\end{array}$ | Heat generated by the GB, $\forall t\in \mathcal{T}$ |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{EB},\mathit{t}}}\end{array}$ | Heat generated by EB |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{PV},\mathit{t}}}\end{array}$ | Power usage of PV at time t (extra power is sold to the grid) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{ST},\mathit{t}}}\end{array}$ | Power usage of ST at time t (extra heat is lost) |

$\begin{array}{c}\hfill {{\mathit{U}}_{\mathit{e},\mathit{t}}}\end{array}$ | Electricity bought from the grid at time t |

$\begin{array}{c}\hfill {{\mathit{V}}_{\mathit{e},\mathit{t}}}\end{array}$ | Electricity sold to the grid at time t |

Parameter | Description |
---|---|

${P}_{CHP,nom}^{min}$, ${P}_{CHP,nom}^{max}$ | Minimum and maximum nominal power of the CHP |

${P}_{GB,nom}^{min}$, ${P}_{GB,nom}^{max}$ | Minimum and maximum nominal power of the GB |

${P}_{EB,nom}^{min}$, ${P}_{EB,nom}^{max}$ | Minimum and maximum nominal power of the EB |

${A}_{PV}^{min}$, ${A}_{PV}^{max}$ | Minimum and maximum area of the PV panels |

${A}_{ST}^{min}$, ${A}_{ST}^{max}$ | Minimum and maximum area of the ST panels |

${L}_{e,t}$ | Electricity demand of the system at time $t\in \mathcal{T}$ |

${L}_{h,t}$ | Heat demand of the system at time $t\in \mathcal{T}$ |

a, b, c | Electrical efficiency coefficients of the CHP |

${\eta}_{th,CHP}$ | Thermal efficiency of the heat recuperation system of the CHP |

${\eta}_{GB}$ | Efficiency of the GB |

${\eta}_{EB}$ | Efficiency of the EB |

${\eta}_{DC/AC}$ | PV inverters’ efficiency |

${\eta}_{ref}$ | Reference efficiency for PV |

${G}_{\beta ,t}$ | Global solar radiation at time $t\in \mathcal{T}$ |

$\alpha $ | Temperature coefficient of PV |

${T}_{ref}$ | Reference temperature |

${T}_{a,t}$ | Outdoor temperature at time $t\in \mathcal{T}$ |

${P}_{panel,nom}$ | Nominal power of a PV panel |

${A}_{panel}$ | Area of a PV panel |

${\eta}_{0}$ | Optical efficiency of ST |

${U}_{loss}$ | Thermal loss coefficient of ST |

${T}_{w,m}$ | Mean water temperature in the ST collector |

${A}_{total}$ | Total area available for solar panels |

${\gamma}_{inv,j}$ | Investment cost for technology $j\in \mathcal{M}$ |

${\gamma}_{O\&M,j,f}$ | Fixed part of the operation cost for technology $j\in \mathcal{M}$ |

${\gamma}_{O\&M,j,v}$ | Variable part of the operation cost for technology $j\in \mathcal{M}$ |

${C}_{gr,t}$ | Cost of electricity bought from the grid at time $t\in \mathcal{T}$ |

${C}_{g}$ | Cost of gas bought |

${I}_{gr}$ | Price of electricity sold to the grid |

Technology j | ${\mathit{C}}_{\mathit{inv},\mathit{j}}$ | ${\mathit{C}}_{\mathit{O}\&\mathit{M},\mathit{j}}$ |
---|---|---|

$\mathit{CHP}$ | ${\gamma}_{inv,CHP}\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{CHP},\mathit{nom}}}\end{array}$ | ${\gamma}_{O\&M,CHP,v}{\sum}_{t\in \mathcal{T}}\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{CHP},\mathit{t}}}\end{array}$ |

$\mathit{GB}$ | ${\gamma}_{inv,GB}\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{GB},\mathit{nom}}}\end{array}$ | ${\gamma}_{O\&M,GB,f}\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{GB},\mathit{nom}}}\end{array}$ |

$\mathit{EB}$ | ${\gamma}_{inv,EB}\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{EB},\mathit{nom}}}\end{array}$ | ${\gamma}_{O\&M,EB,f}\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{EB},\mathit{nom}}}\end{array}+{\gamma}_{O\&M,EB,v}{\sum}_{t\in \mathcal{T}}\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{EB},\mathit{t}}}\end{array}$ |

$\mathit{PV}$ | ${\gamma}_{inv,PV}{P}_{panel,nom}\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{PV}}}\end{array}/{A}_{panel}$ | ${\gamma}_{O\&M,PV,f}{P}_{panel,nom}\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{PV}}}\end{array}/{A}_{panel}$ |

$\mathit{ST}$ | ${\gamma}_{inv,ST}\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{ST}}}\end{array}$ | ${\gamma}_{O\&M,ST,f}\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{ST}}}\end{array}$ |

max $\phantom{\rule{1.em}{0ex}}ATCR=100\ast (1-\frac{{\mathit{ATC}}_{\mathit{MES}}}{\mathit{A}\mathit{T}{\mathit{C}}_{\mathit{r}\mathit{e}\mathit{f}}})$ | (3) | |

max $\phantom{\rule{1.em}{0ex}}{\tau}_{RES}=100\ast \frac{{\sum}_{t\in \mathcal{T}}(\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{PV},\mathit{t}}}\end{array}+\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{ST},\mathit{t}}}\end{array})}{{\sum}_{t\in \mathcal{T}}({L}_{e,t}+{L}_{h,t})}$ | (4) | |

s.t. | ||

$\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{PV}}}\end{array}+\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{ST}}}\end{array}\le {A}_{total}$ | (5) | |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{CHP},\mathit{t}}}\end{array}\le \begin{array}{c}\hfill {{\mathit{P}}_{\mathit{CHP},\mathit{nom}}}\end{array}$ | $\forall t\in \mathcal{T}$ | (6) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{GB},\mathit{t}}}\end{array}\le \begin{array}{c}\hfill {{\mathit{P}}_{\mathit{GB},\mathit{nom}}}\end{array}$ | $\forall t\in \mathcal{T}$ | (7) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{EB},\mathit{t}}}\end{array}\le \begin{array}{c}\hfill {{\mathit{P}}_{\mathit{EB},\mathit{nom}}}\end{array}$ | $\forall t\in \mathcal{T}$ | (8) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{CHP},\mathit{t}}}\end{array}=(a+b\left(\frac{\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{CHP},\mathit{t}}}\end{array}}{\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{CHP},\mathit{nom}}}\end{array}}\right)+c{\left(\frac{\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{CHP},\mathit{t}}}\end{array}}{\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{CHP},\mathit{nom}}}\end{array}}\right)}^{2})\begin{array}{c}\hfill {{\mathit{F}}_{\mathit{g},\mathit{CHP},\mathit{t}}}\end{array}$ | $\forall t\in \mathcal{T}$ | (9) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{CHP},\mathit{t}}}\end{array}\le {\eta}_{th,CHP}(\begin{array}{c}\hfill {{\mathit{F}}_{\mathit{g},\mathit{CHP},\mathit{t}}}\end{array}-\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{CHP},\mathit{t}}}\end{array})$ | $\forall t\in \mathcal{T}$ | (12) |

$\begin{array}{c}\hfill {{\mathit{V}}_{\mathit{e},\mathit{t}}}\end{array}+\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{PV},\mathit{t}}}\end{array}=\begin{array}{c}\hfill {{\mathit{A}}_{\mathit{PV}}}\end{array}{\eta}_{DC/AC}{\eta}_{ref}(1-\alpha (30+0.0175({G}_{\beta ,t}-300)$ | ||

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+1.14({T}_{a,t}-25)-{T}_{ref}\left)\right){G}_{\beta ,t}$ | $\forall t\in \mathcal{T}$ | (13) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{ST},\mathit{t}}}\end{array}\le \begin{array}{c}\hfill {{\mathit{A}}_{\mathit{ST}}}\end{array}({G}_{\beta ,t}{\eta}_{0}-{U}_{loss}({T}_{w,m}-{T}_{a,t}))$ | $\forall t\in \mathcal{T}$ | (16) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{CHP},\mathit{t}}}\end{array}+\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{e},\mathit{PV},\mathit{t}}}\end{array}+\begin{array}{c}\hfill {{\mathit{U}}_{\mathit{e},\mathit{t}}}\end{array}-\frac{\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{EB},\mathit{t}}}\end{array}}{{\eta}_{EB}}={L}_{e,t}$ | $\forall t\in \mathcal{T}$ | (17) |

$\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{CHP},\mathit{t}}}\end{array}+\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{GB},\mathit{t}}}\end{array}+\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{EB},\mathit{t}}}\end{array}+\begin{array}{c}\hfill {{\mathit{P}}_{\mathit{h},\mathit{ST},\mathit{t}}}\end{array}={L}_{h,t}$ | $\forall t\in \mathcal{T}$ | (18) |

Technology j | ${\mathit{\gamma}}_{\mathit{inv},\mathit{j}}$ | ${\mathit{\gamma}}_{\mathit{O}\&\mathit{M},\mathit{j},\mathit{f}}$ | ${\mathit{\gamma}}_{\mathit{O}\&\mathit{M},\mathit{j},\mathit{v}}$ |
---|---|---|---|

$\mathit{CHP}$ | 1140 €/$\mathrm{k}$${\mathrm{W}}_{\mathrm{e}}$ | - | 21 €/$\mathrm{M}$${\mathrm{Wh}}_{\mathrm{e}}$ |

$\mathit{GB}$ | 90 €/$\mathrm{k}$${\mathrm{W}}_{\mathrm{th}}$ | 3.15 €/$\mathrm{k}$${\mathrm{W}}_{\mathrm{th}}$$\mathrm{year}$ | - |

$\mathit{EB}$ | 100 €/$\mathrm{k}$${\mathrm{W}}_{\mathrm{th}}$ | 1 €/$\mathrm{k}$${\mathrm{W}}_{\mathrm{th}}$$\mathrm{year}$ | 0.8 €/$\mathrm{M}$${\mathrm{Wh}}_{\mathrm{th}}$ |

$\mathit{PV}$ | 1000 €/$\mathrm{k}$${\mathrm{W}}_{\mathrm{e}}$ | 15 €/$\mathrm{k}$${\mathrm{W}}_{\mathrm{e}}$$\mathrm{year}$ | - |

$\mathit{ST}$ | 615 €/$\mathrm{m}$${}^{2}$ | 10 €/$\mathrm{m}$${}^{2}$$\mathrm{year}$ | - |

Parameter | Description |
---|---|

${P}_{CHP,nom}^{min}=100\mathrm{k}{\mathrm{W}}_{\mathrm{e}}$, ${P}_{CHP,nom}^{max}=1000\mathrm{k}{\mathrm{W}}_{\mathrm{e}}$ | Minimum and maximum nominal power of the CHP |

${P}_{GB,nom}^{min}=100\mathrm{k}{\mathrm{W}}_{\mathrm{th}}$, ${P}_{GB,nom}^{max}=3000\mathrm{k}{\mathrm{W}}_{\mathrm{th}}$ | Minimum and maximum nominal power of the GB |

${P}_{EB,nom}^{min}=100\mathrm{k}{\mathrm{W}}_{\mathrm{th}}$, ${P}_{EB,nom}^{max}=3000\mathrm{k}{\mathrm{W}}_{\mathrm{th}}$ | Minimum and maximum nominal power of the EB |

${A}_{PV}^{min}=0{\mathrm{m}}^{2}$, ${A}_{PV}^{max}=\text{10,000}{\mathrm{m}}^{2}$ | Minimum and maximum area of the PV panels |

${A}_{ST}^{min}=0{\mathrm{m}}^{2}$, ${A}_{ST}^{max}=\text{10,000}{\mathrm{m}}^{2}$ | Minimum and maximum area of the ST panels |

${L}_{e,t}$ ($\mathrm{k}{\mathrm{W}}_{\mathrm{e}}$) | Electricity demand of the system |

${L}_{h,t}$ ($\mathrm{k}{\mathrm{W}}_{\mathrm{th}}$) | Heat demand of the system |

$a=0.1$, $b=0.4$, $c=-0.2$ | Electrical efficiency coefficients of the CHP |

$\beta =0.3$ | Constant CHP electrical efficiency |

${\eta}_{th,CHP}=0.8$ | Thermal efficiency of the heat recuperation system of the CHP |

${\eta}_{GB}=0.8$ | Efficiency of the GB |

${\eta}_{EB}=0.8$ | Efficiency of the EB |

${\eta}_{DC/AC}=0.9$ | PV inverters’ efficiency |

${\eta}_{ref}=0.155$ | Reference efficiency for PV |

${G}_{\beta ,t}$ ($\mathrm{W}/{\mathrm{m}}^{-2}$) | Global solar radiation |

$\alpha =0.43\%{/}^{\circ}\mathrm{C}$ | Temperature coefficient of PV |

${T}_{ref}=25{}^{\circ}\mathrm{C}$ | Reference temperature |

${T}_{a,t}$ (${}^{\circ}\mathrm{C}$) | Outdoor temperature |

${P}_{panel,nom}=250\mathrm{W}$ | Nominal power of a PV panel |

${A}_{panel}=1.6{\mathrm{m}}^{2}$ | Area of a PV panel |

${\eta}_{0}=0.8$ | Optical efficiency of ST |

${U}_{loss}=5\mathrm{W}/{\mathrm{m}}^{-2}{/}^{\circ}{\mathrm{C}}^{-1}$ | Thermal loss coefficient of ST |

${T}_{w,m}=45{}^{\circ}\mathrm{C}$ | Mean water temperature in the ST collector |

${A}_{total}=\text{10,000}{\mathrm{m}}^{2}$ | Total area available for solar panels |

${C}_{gr,t}$ = 0.13 €/$\mathrm{k}$$\mathrm{Wh}$ | Cost of electricity bought from the grid (0 h–7 h) |

${C}_{gr,t}$ = 0.17 €/$\mathrm{k}$$\mathrm{Wh}$ | Cost of electricity bought from the grid (8 h–23 h) |

${C}_{g}$ = 0.076 €/$\mathrm{k}$$\mathrm{Wh}$ | Cost of gas bought |

${I}_{gr}$ = 0.1 €/$\mathrm{k}$$\mathrm{Wh}$ | Price of electricity sold to the grid |

${\mathit{L}}_{\mathit{e},\mathit{t}}$ | ${\mathit{L}}_{\mathit{h},\mathit{t}}$ | |
---|---|---|

Summer | $\mu =322\mathrm{kW}$, $\sigma =54\mathrm{kW}$ | $\mu =55\mathrm{kW}$, $\sigma =58\mathrm{kW}$ |

Mid-season | $\mu =337\mathrm{kW}$, $\sigma =55\mathrm{kW}$ | $\mu =395\mathrm{kW}$, $\sigma =390\mathrm{kW}$ |

Winter | $\mu =526\mathrm{kW}$, $\sigma =82\mathrm{kW}$ | $\mu =1459\mathrm{kW}$, $\sigma =528\mathrm{kW}$ |

${\mathit{T}}_{\mathit{t}}$ | ${\mathit{G}}_{\mathit{\beta},\mathit{t}}$ | |

Summer | $\mu =15.1\xb0\mathrm{C}$, $\sigma =2.8\xb0\mathrm{C}$ | $\mu =175\mathrm{W}{\mathrm{m}}^{-2}$, $\sigma =214\mathrm{W}{\mathrm{m}}^{-2}$ |

Mid-season | $\mu =16.5\xb0\mathrm{C}$, $\sigma =4.5\xb0\mathrm{C}$ | $\mu =196\mathrm{W}{\mathrm{m}}^{-2}$, $\sigma =279\mathrm{W}{\mathrm{m}}^{-2}$ |

Winter | $\mu =5.4\xb0\mathrm{C}$, $\sigma =4.6\xb0\mathrm{C}$ | $\mu =123\mathrm{W}{\mathrm{m}}^{-2}$, $\sigma =224\mathrm{W}{\mathrm{m}}^{-2}$ |

Run | ${\mathit{P}}_{\mathit{e},\mathit{CHP}}$ | ${\mathit{P}}_{\mathit{e},\mathit{PV}}$ | ${\mathit{U}}_{\mathit{e}}$ | ${\mathit{P}}_{\mathit{h},\mathit{CHP}}$ | ${\mathit{P}}_{\mathit{h},\mathit{GB}}$ | ${\mathit{P}}_{\mathit{h},\mathit{EB}}$ | ${\mathit{P}}_{\mathit{h},\mathit{ST}}$ | $\mathit{ATCR}$ | ${\mathit{\tau}}_{\mathit{RES}}$ |
---|---|---|---|---|---|---|---|---|---|

Summer (1) | 7% | $49.6\%$ | $43.4\%$ | $99.9\%$ | $0.1\%$ | 0% | 0% | 34.47 | 42.43 |

Summer (6) | $6.4\%$ | $50.2\%$ | $43.4\%$ | $94.4\%$ | $5.5\%$ | $0.1\%$ | 0% | 34.46 | 42.91 |

Summer (10) | $6.3\%$ | $50.4\%$ | $43.2\%$ | $92.7\%$ | $5.3\%$ | $1.9\%$ | 0% | 34.41 | 43.29 |

Mid-season (1) | $42.2\%$ | $41.5\%$ | $16.2\%$ | $69.3\%$ | $30.5\%$ | $0.2\%$ | 0% | 27.57 | 18.87 |

Mid-season (6) | $41.1\%$ | $42.7\%$ | $16.2\%$ | $67.2\%$ | $32.5\%$ | $0.2\%$ | 0% | 27.55 | 19.40 |

Mid-season (10) | $40.6\%$ | $43.3\%$ | $16.1\%$ | $67.2\%$ | 32% | $0.8\%$ | 0% | 27.45 | 19.83 |

Winter (1) | $93.8\%$ | $5.2\%$ | 1% | 65% | 35% | 0% | 0% | 12.26 | 1.38 |

Winter (6) | $71.4\%$ | $27.6\%$ | 1% | 52% | $47.5\%$ | $0.5\%$ | 0% | 12.01 | 7.44 |

Winter (10) | $77.9\%$ | $18.6\%$ | $3.5\%$ | $55.2\%$ | $34.7\%$ | $0.1\%$ | 10% | 3.14 | 12.29 |

Run | ${\mathit{P}}_{\mathit{e},\mathit{CHP}}$ | ${\mathit{P}}_{\mathit{e},\mathit{PV}}$ | ${\mathit{U}}_{\mathit{e}}$ | ${\mathit{V}}_{\mathit{e}}$ | ${\mathit{P}}_{\mathit{h},\mathit{CHP}}$ | ${\mathit{P}}_{\mathit{h},\mathit{GB}}$ | ${\mathit{P}}_{\mathit{h},\mathit{EB}}$ | ${\mathit{P}}_{\mathit{h},\mathit{ST}}$ | $\mathit{ATCR}$ | ${\mathit{\tau}}_{\mathit{RES}}$ |
---|---|---|---|---|---|---|---|---|---|---|

Year (1) | 2.02 GWh | 840 MWh | 701 MWh | 1.14 GWh | 4.05 GWh | 2.02 GWh | 1.54 MWh | 0 Wh | 15.71 | 8.73 |

Year (6) | 1.69 GWh | 1.22 GWh | 698 MWh | 755 MWh | 3.44 GWh | 2.59 GWh | 45 MWh | 0 Wh | 15.51 | 12.68 |

Year (10) | 1.75 GWh | 1.2 GWh | 769 MWh | 310 MWh | 3.55 GWh | 2.07 GWh | 129 MWh | 322 MWh | 4.48 | 15.84 |

Run | ${\mathit{P}}_{\mathit{e},\mathit{CHP}}$ | ${\mathit{P}}_{\mathit{e},\mathit{PV}}$ | ${\mathit{U}}_{\mathit{e}}$ | ${\mathit{V}}_{\mathit{e}}$ | ${\mathit{P}}_{\mathit{h},\mathit{CHP}}$ | ${\mathit{P}}_{\mathit{h},\mathit{GB}}$ | ${\mathit{P}}_{\mathit{h},\mathit{EB}}$ | ${\mathit{P}}_{\mathit{h},\mathit{ST}}$ | $\mathit{ATCR}$ | ${\mathit{\tau}}_{\mathit{RES}}$ |
---|---|---|---|---|---|---|---|---|---|---|

Year (1) | 2.22 GWh | 760 MWh | 576 MWh | 1.22 GWh | 4.15 GWh | 1.91 GWh | 1.31 MWh | 0 Wh | 16.14 | 7.89 |

Year (6) | 1.82 GWh | 1.18 GWh | 578 MWh | 791 MWh | 3.39 GWh | 2.66 GWh | 16.4 MWh | 0 Wh | 16.00 | 12.31 |

Year (10) | 1.88 GWh | 1.19 GWh | 642 MWh | 299 MWh | 3.51 GWh | 2.1 GWh | 124 MWh | 335 MWh | 4.32 | 15.84 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mallégol, A.; Khannoussi, A.; Mohammadi, M.; Lacarrière, B.; Meyer, P.
Handling Non-Linearities in Modelling the Optimal Design and Operation of a Multi-Energy System. *Mathematics* **2023**, *11*, 4855.
https://doi.org/10.3390/math11234855

**AMA Style**

Mallégol A, Khannoussi A, Mohammadi M, Lacarrière B, Meyer P.
Handling Non-Linearities in Modelling the Optimal Design and Operation of a Multi-Energy System. *Mathematics*. 2023; 11(23):4855.
https://doi.org/10.3390/math11234855

**Chicago/Turabian Style**

Mallégol, Antoine, Arwa Khannoussi, Mehrdad Mohammadi, Bruno Lacarrière, and Patrick Meyer.
2023. "Handling Non-Linearities in Modelling the Optimal Design and Operation of a Multi-Energy System" *Mathematics* 11, no. 23: 4855.
https://doi.org/10.3390/math11234855