Abstract
In this work, a new tri-level stochastic optimization formulation is developed for the capacity expansion (CE) planning of an integrated multi-carrier energy system comprising electricity, natural gas, and district heating (DH) networks that explicitly captures the trade-off between economic optimality and system resilience in the face of extreme events. When prior works on the (dis)integration of energy vectors often consider energy vectors independently or overlook disrupted system recovery dynamics, the present research develops an integrated multi-carrier framework that simultaneously determines long-term investment decisions, short-term operational dispatch, and emergency reconfiguration measures. The problem is formulated in a hierarchical manner: the upper level searches for optimal capacity expansion for lines, pipelines, and conversion technologies (CHP, P2G, heat pumps), the middle level solves minimum operational cost problem under ordinary stochastic conditions, and the lower-level deals with maximum load restoration in N-k contingencies by a quantitative resiliency measure. To make the difficult mixed-integer problem computationally tractable, we employ an iterative Column-and-Constraint Generation (C&CG) algorithm. A numerical example of a coupled test system over a ten-year time period displays the feasibility of the presented approach. It is shown that the combined resilience-oriented design planning decreases the total expected costs as well as the resilience cost penalty by ca. 8.4% over the single-energy-system design and even surpasses the performance of deterministic methods considering system resilience, although with increasing costs. Critically, the framework features an endogenous, quantifiable resilience metric that is co-optimized with economic objectives, transforming resilience from a passive design constraint into an active driver of investment decisions. In addition, the framework enables a significant increase in renewables penetration (from18% to 54%) through exploiting multi-carrier flexibility to buffer intermittency. These results can help guide policy makers on the importance of cross-sectoral coupling to increase infrastructure robustness and demonstrate that such resilience-focused investments are instrumental in enabling deep decarbonization of the energy system.
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Abbreviations
- \(\mathcal{T}\) :
-
Set of planning years, indexed by \(t\)
- \(\mathcal{S}\) :
-
Set of seasonal periods, indexed by \(s\)
- \(\mathcal{H}\) :
-
Set of hourly time steps, indexed by \(h\)
- \({\mathcal{N}}^{e}\) :
-
Set of electrical buses, indexed by \(i,j\)
- \({\mathcal{N}}^{g}\) :
-
Set of gas nodes, indexed by \(m,n\)
- \({\mathcal{N}}^{h}\) :
-
Set of heating nodes, indexed by \(p,q\)
- \({\mathcal{L}}^{e}\) :
-
Set of electrical transmission lines, indexed by \({l}^{e}\)
- \({\mathcal{L}}^{g}\) :
-
Set of gas pipelines, indexed by \({l}^{g}\)
- \({\mathcal{L}}^{h}\) :
-
Set of heating pipes, indexed by \({l}^{h}\)
- \({\mathcal{G}}^{conv}\) :
-
Set of conventional generators, indexed by \(g\)
- \({\mathcal{G}}^{wind}\) :
-
Set of wind farms, indexed by \(w\)
- \({\mathcal{G}}^{solar}\) :
-
Set of solar installations, indexed by \(pv\)
- \({\mathcal{G}}^{chp}\) :
-
Set of combined heat and power (CHP) units, indexed by \(c\)
- \({\mathcal{G}}^{gas}\) :
-
Set of gas-fired generators
- \({\mathcal{U}}^{p2g}\) :
-
Set of power-to-gas facilities, indexed by \(k\)
- \({\mathcal{U}}^{boiler}\) :
-
Set of gas boilers, indexed by \(b\)
- \({\mathcal{U}}^{hp}\) :
-
Set of heat pumps, indexed by \(hp\)
- \({\mathcal{U}}^{storage,e}\) :
-
Set of electrical storage systems, indexed by \(es\)
- \({\mathcal{U}}^{storage,g}\) :
-
Set of gas storage facilities, indexed by \(gs\).
- \({\mathcal{U}}^{storage,h}\) :
-
Set of thermal storage units, indexed by \(hs\)
- \(\Omega\) :
-
Set of operational scenarios, indexed by \(\omega\)
- \(\Xi\) :
-
Set of disruption scenarios, indexed by \(\xi\)
- \(\mathcal{C}\) :
-
Set of candidate investment options, indexed by \(\iota\)
- \({\mathcal{C}}^{harden}\) :
-
Subset of candidate hardening investments
- \(\mathcal{Z}\) :
-
Set of network zones for resilience assessment, indexed by \(z\)
- \({\mathcal{Z}}^{critical}\) :
-
Set of critical zones
- \({\mathcal{Z}}^{microgrid}\) :
-
Set of microgrid zones
- \({\mathcal{M}}^{supply}\) :
-
Set of gas supply nodes
- \({\mathcal{N}}^{critical,e}\) :
-
Set of electrical buses with critical loads
- \({\mathcal{P}}_{i,j}^{path}\) :
-
Set of lines in a path connecting buses \(i\) and \(j\)
- \({\mathcal{C}}^{cut}\) :
-
A cutset in the electrical network
- \({\mathcal{N}}^{island}\) :
-
Set of buses forming an island
- \({C}_{\iota }^{inv}\) :
-
Investment cost for candidate option \(\iota\)
- \({C}_{g,t}^{fuel}\) :
-
Fuel cost for generator \(g\) in year \(t\)
- \({C}_{g}^{startup}\) :
-
Startup cost for generator \(g\)
- \({C}_{g}^{shutdown}\) :
-
Shutdown cost for generator \(g\)
- \({C}_{m,t}^{gas}\) :
-
Cost of gas supply at node \(m\) in year \(t\)
- \({C}_{c,t}^{fuel,chp}\) :
-
Fuel cost for CHP unit \(c\)
- \({C}_{w}^{curtail,wind}\) :
-
Penalty for wind power curtailment
- \({C}_{pv}^{curtail,solar}\) :
-
Penalty for solar power curtailment
- \({C}_{es}^{degrade}\) :
-
Degradation cost for electrical storage \(es\)
- \({C}_{gs}^{operate}\) :
-
Operating cost for gas storage \(gs\)
- \({C}_{t,s,h}^{curtail,e}\) :
-
Penalty for electrical load curtailment
- \({C}_{t,s,h}^{curtail,g}\) :
-
Penalty for gas load curtailment
- \({C}_{t,s,h}^{curtail,h}\) :
-
Penalty for heating load curtailment
- \({C}_{{l}^{e}}^{switch}\) :
-
Cost of switching electrical line \({l}^{e}\)
- \({C}_{i}^{blackstart}\) :
-
Cost associated with black start capability at bus \(i\)
- \({C}_{z}^{resilience}\) :
-
Resilience cost for zone \(z\)
- \({C}_{g}^{attack}\) :
-
Cost for attacker to disable generator \(g\)
- \({C}_{{l}^{e}}^{attack}\) :
-
Cost for attacker to disable line \({l}^{e}\)
- \({\overline{P}}_{g}\) :
-
Maximum power output of generator \(g\)
- \({\underline{P}}_{g}\) :
-
Minimum power output of generator \(g\)
- \({R}_{g}^{up}\) :
-
Ramp-up rate of generator \(g\)
- \({R}_{g}^{down}\) :
-
Ramp-down rate of generator \(g\)
- \({T}_{g}^{up}\) :
-
Minimum up time of generator \(g\)
- \({T}_{g}^{down}\) :
-
Minimum down time of generator \(g\)
- \({\eta }_{g}^{heat}\) :
-
Heat rate of generator \(g\)
- \({\eta }_{c}^{elec}\) :
-
Electrical efficiency of CHP unit \(c\)
- \({\eta }_{c}^{heat}\) :
-
Thermal efficiency of CHP unit \(c\)
- \({\overline{\Gamma }}_{c}^{H2P}\),\({\underset{\_}{\Gamma }}_{c}^{H2P}\) :
-
Maximum and minimum heat-to-power ratio for CHP unit \(c\)
- \({\eta }_{k}^{p2g}\) :
-
Efficiency of power-to-gas facility \(k\)
- \({\eta }_{b}^{boiler}\) :
-
Efficiency of gas boiler \(b\)
- \({COP}_{hp}\) :
-
Coefficient of performance for heat pump \(hp\)
- \({\eta }_{es}^{charge}\) :
-
Charging efficiency of electrical storage \(es\)
- \({\eta }_{es}^{discharge}\) :
-
Discharging efficiency of electrical storage \(es\)
- \({\rho }_{es}^{decay}\) :
-
Self-discharge rate of electrical storage \(es\)
- \({\varrho }_{es}^{E2P}\) :
-
Energy-to-power ratio for electrical storage \(es\)
- \({\overline{F}}_{{l}^{e}}\) :
-
Capacity of electrical line \({l}^{e}\)
- \({X}_{{l}^{e}}\) :
-
Reactance of electrical line \({l}^{e}\)
- \({{\ell}}_{{l}^{e}}^{loss}\) :
-
Fractional transmission loss on line \({l}^{e}\)
- \({\overline{Q}}_{{l}^{g}}\) :
-
Capacity of gas pipeline \({l}^{g}\)
- \({K}_{{l}^{g}}\) :
-
Pipeline constant for gas pipeline \({l}^{g}\)
- \({\overline{H}}_{{l}^{h}}\) :
-
Capacity of heating pipe \({l}^{h}\)
- \({\Lambda }_{{l}^{h}}\) :
-
Thermal loss coefficient for heating pipe \({l}^{h}\)
- \(\underset{m}{\stackrel{g,min}{\text{Pr}}}\),\(\underset{m}{\stackrel{g,max}{\text{Pr}}}\) :
-
Minimum and maximum gas pressure at node \(m\)
- \({T}_{p}^{h,min}\),\({T}_{p}^{h,max}\) :
-
Minimum and maximum temperature at heating node \(p\)
- \({\overline{E}}_{es}\) :
-
Energy capacity of electrical storage \(es\)
- \({\overline{P}}_{es}^{ch}\),\({\overline{P}}_{es}^{dis}\) :
-
Charging/Discharging power limit of electrical storage \(es\)
- \({\overline{V}}_{gs}\),\({\underline{V}}_{gs}\) :
-
Maximum and minimum volume capacity of gas storage \(gs\)
- \({\overline{Q}}_{gs}^{inj}\),\({\overline{Q}}_{gs}^{with}\) :
-
Injection/Withdrawal rate limit of gas storage \(gs\)
- \({\overline{Q}}_{hs}\) :
-
Capacity of thermal storage \(hs\)
- \({P}_{i,t,s,h,\omega }^{d,e}\) :
-
Electrical demand at bus \(i\)
- \({Q}_{m,t,s,h,\omega }^{d,g}\) :
-
Gas demand at node \(m\)
- \({H}_{p,t,s,h,\omega }^{d,h}\) :
-
Heating demand at node \(p\)
- \({\overline{P}}_{w}^{wind}\) :
-
Capacity of existing wind farm \(w\)
- \({\overline{P}}_{pv}^{solar}\) :
-
Capacity of existing solar installation \(pv\)
- \({\overline{P}}_{c}^{chp}\) :
-
Electrical capacity of CHP unit \(c\)
- \({\overline{Q}}_{c}^{chp}\) :
-
Thermal capacity of CHP unit \(c\)
- \({\overline{Q}}_{b}^{boiler}\) :
-
Capacity of gas boiler \(b\)
- \({\overline{P}}_{hp}^{hp}\) :
-
Electrical capacity of heat pump \(hp\)
- \({\overline{Q}}_{hp}^{hp}\) :
-
Thermal capacity of heat pump \(hp\)
- \({\overline{P}}_{k}^{p2g}\) :
-
Capacity of power-to-gas facility \(k\)
- \({Q}_{m}^{supply,base}\) :
-
Base gas supply capacity at node \(m\)
- \({p}_{w,t,s,h,\omega }^{avail,wind}\) :
-
Available wind power generation
- \({p}_{pv,t,s,h,\omega }^{avail,solar}\) :
-
Available solar power generation
- \({\overline{\kappa }}_{{l}^{e}}^{max,e}\),\({\overline{\kappa }}_{{l}^{e}}^{total,e}\) :
-
Maximum annual and total capacity expansion for electrical line \({l}^{e}\)
- \({\overline{\kappa }}_{{l}^{g}}^{max,g}\),\({\overline{\kappa }}_{{l}^{g}}^{total,g}\) :
-
Maximum annual and total capacity expansion for gas pipeline \({l}^{g}\)
- \({\underset{\_}{\kappa }}_{g}^{min,gen}\),\({\overline{\kappa }}_{g}^{max,gen}\) :
-
Minimum and maximum capacity expansion for generator \(g\)
- \({\overline{P}}_{g}^{total}\) :
-
Technical maximum capacity for generator \(g\)
- \({B}_{t}^{budget}\) :
-
Available investment budget for year \(t\)
- \({N}_{i}^{min,connect}\) :
-
Minimum number of connections required for bus \(i\)
- \({\overline{Q}}_{m}^{inject,max}\) :
-
Maximum allowable gas injection at node \(m\)
- \({RM}^{e}\), \({RM}^{g}\),\({RM}^{h}\) :
-
Electrical, gas, and heating planning reserve margins
- \({\alpha }^{wind}\) :
-
Capacity credit factor for wind generation
- \({\beta }_{t}^{RES}\) :
-
Minimum renewable penetration ratio in year \(t\)
- \({\gamma }_{z}^{gen}\) :
-
Minimum local generation ratio for zone \(z\)
- \({\chi }_{t}^{harden,min}\) :
-
Minimum fraction of investment allocated to hardening
- \({\varsigma }^{reserve,g}\) :
-
Required gas reserve duration (hours)
- \({\varepsilon }^{p2g,min}\) :
-
Minimum power-to-gas capacity ratio relative to wind capacity
- \({N}_{z}^{lines,total}\) :
-
Total number of lines in zone \(z\)
- \({\pi }_{\omega }\) :
-
Probability of operational scenario \(\omega\)
- \({\pi }_{\xi }\) :
-
Probability of disruption scenario \(\xi\)
- \({A}_{{l}^{e},\xi }\) :
-
Availability of electrical line \({l}^{e}\) under disruption \(\xi\)
- \({A}_{{l}^{g},\xi }\) :
-
Availability of gas pipeline \({l}^{g}\)
- \({A}_{g,\xi }\) :
-
Availability of generator \(g\)
- \({A}_{m,\xi }^{supply}\) :
-
Availability of gas supply at node \(m\)
- \({\Gamma }_{\xi }^{duration}\) :
-
Duration of disruption scenario \(\xi\)
- \({\Psi }_{z}^{critical}\) :
-
Criticality weight of zone \(z\)
- \({P}^{min,island}\) :
-
Minimum generation required to form an energized island.
- \({\alpha }_{i}^{critical}\) :
-
Criticality factor for bus \(i\)
- \({\Delta }_{i}^{max,curtail}\) :
-
Maximum allowable cumulative electrical load curtailment
- \({\Phi }^{min,resil}\) :
-
Minimum acceptable system resilience
- \({\epsilon }^{carrier,balance}\) :
-
Maximum allowable imbalance between energy carrier curtailments
- \({A}_{{l}^{e},\xi }^{base}\) :
-
Baseline availability of component \({l}^{e}\)
- \({\beta }_{z}^{microgrid}\) :
-
Resilience enhancement factor from microgrid capability
- \({B}^{attack}\) :
-
Attacker’s resource budget
- \({R}_{t,s,h}^{spin,up}\),\({R}_{t,s,h}^{spin,down}\) :
-
Upward and downward spinning reserve requirements
- \({\delta }_{t}\) :
-
Discount factor for year \(t\)
- \({D}_{s}\) :
-
Duration of season \(s\) in days
- \({\eta }_{h}\) :
-
Weight of hour \(h\) within seasonal representation
- \({M}^{big}\) :
-
A sufficiently large constant (Big-M)
- \({x}_{\iota ,t}\) :
-
Binary investment decision for candidate option \(\iota\) in year \(t\)
- \({y}_{{l}^{e},t}^{inv,e}\) :
-
Binary decision to invest in electrical line \({l}^{e}\)
- \({y}_{{l}^{e},t}^{inv,e}\) :
-
Binary decision to invest in gas pipeline \({l}^{g}\)
- \({y}_{g,t}^{inv,gen}\) :
-
Binary decision to invest in generator \(g\)
- \({y}_{c,t}^{inv,chp}\) :
-
Binary decision to invest in CHP unit \(c\)
- \({y}_{{l}^{e},t}^{harden,line}\) :
-
Binary hardening decision for electrical line \({l}^{e}\)
- \({\kappa }_{{l}^{e},t}^{cap,e}\) :
-
Continuous capacity expansion for electrical line \({l}^{e}\)
- \({\kappa }_{{l}^{g},t}^{cap,g}\) :
-
Capacity expansion for gas pipeline \({l}^{g}\)
- \({\kappa }_{g,t}^{cap,gen}\) :
-
Capacity expansion for generator \(g\)
- \({\kappa }_{w,t}^{cap,wind}\) :
-
Capacity expansion for wind farm \(w\)
- \({\kappa }_{pv,t}^{cap,solar}\) :
-
Capacity expansion for solar installation \(pv\)
- \({\kappa }_{c,t}^{elec,chp}\) , \({\kappa }_{c,t}^{heat,chp}\) :
-
Electrical and thermal capacity expansion for CHP unit \(c\)
- \({\kappa }_{m,t}^{supply,g}\) :
-
Gas supply capacity expansion at node \(m\)
- \({\kappa }_{k,t}^{cap,p2g}\) :
-
Capacity expansion for power-to-gas facility \(k\)
- \({\kappa }_{b,t}^{cap,boiler}\) :
-
Capacity expansion for gas boiler \(b\)
- \({\kappa }_{hp,t}^{cap,hp}\) :
-
Capacity expansion for heat pump \(hp\)
- \({\kappa }_{es,t}^{power,es}\) , \({\kappa }_{es,t}^{energy,es}\) :
-
Power and energy capacity expansion for electrical storage \(es\)
- \({\kappa }_{gs,t}^{cap,gs}\) :
-
Capacity expansion for gas storage \(gs\)
- \({\kappa }_{{l}^{h},t}^{cap,h}\) :
-
Capacity expansion for heating pipe \({l}^{h}\)
- \({p}_{g,t,s,h,\omega }^{gen}\) :
-
Power output of conventional generator \(g\)
- \({u}_{g,t,s,h,\omega }^{gen}\) :
-
Binary commitment status of generator \(g\)
- \({v}_{g,t,s,h,\omega }^{startup}\) , \({v}_{g,t,s,h,\omega }^{shutdown}\) :
-
Binary startup/shutdown variables for generator \(g\)
- \({p}_{c,t,s,h,\omega }^{chp,e}\) :
-
Electrical output of CHP unit \(c\)
- \({q}_{c,t,s,h,\omega }^{chp,h}\) :
-
Heat output of CHP unit \(c\)
- \({p}_{w,t,s,h,\omega }^{wind}\) :
-
Power output of wind farm \(w\)
- \({p}_{pv,t,s,h,\omega }^{solar}\) :
-
Power output of solar installation \(pv\)
- \({p}_{k,t,s,h,\omega }^{p2g}\) :
-
Power consumption of power-to-gas facility \(k\)
- \({q}_{k,t,s,h,\omega }^{p2g,out}\) :
-
Gas production from power-to-gas facility \(k\)
- \({f}_{{l}^{e},t,s,h,\omega }^{e}\) :
-
Power flow on electrical line \({l}^{e}\)
- \({\theta }_{i,t,s,h,\omega }^{e}\) :
-
Voltage angle at electrical bus \(i\)
- \({q}_{{l}^{g},t,s,h,\omega }^{g}\) :
-
Gas flow in pipeline \({l}^{g}\)
- \({pr}_{m,t,s,h,\omega }^{g}\) :
-
Gas pressure at node \(m\)
- \({h}_{{l}^{h},t,s,h,\omega }^{h}\) :
-
Heat flow in heating pipe \({l}^{h}\)
- \({\tau }_{p,t,s,h,\omega }^{h}\) :
-
Temperature at heating node \(p\)
- \({q}_{m,t,s,h,\omega }^{supply}\) :
-
Gas supply from external source at node \(m\)
- \({p}_{es,t,s,h,\omega }^{ch}\) , \({p}_{es,t,s,h,\omega }^{dis}\) :
-
Charging/Discharging power of electrical storage \(es\)
- \({e}_{es,t,s,h,\omega }^{soc}\) :
-
State of charge of electrical storage \(es\)
- \({q}_{gs,t,s,h,\omega }^{inj}\) , \({q}_{gs,t,s,h,\omega }^{with}\) :
-
Gas injection/withdrawal for gas storage \(gs\)
- \({v}_{gs,t,s,h,\omega }^{level}\) :
-
Storage level of gas storage \(gs\)
- \({q}_{b,t,s,h,\omega }^{boiler,in}\) :
-
Gas input to boiler \(b\)
- \({q}_{b,t,s,h,\omega }^{boiler,out}\) :
-
Heat output from boiler \(b\)
- \({p}_{hp,t,s,h,\omega }^{hp}\) :
-
Electricity consumption of heat pump \(hp\)
- \({q}_{hp,t,s,h,\omega }^{hp,out}\) :
-
Heat output from heat pump \(hp\)
- \({q}_{hs,t,s,h,\omega }^{ch,h}\) , \({q}_{hs,t,s,h,\omega }^{dis,h}\) :
-
Charging/Discharging heat for thermal storage \(hs\)
- \({f}_{c,t,s,h,\omega }^{chp,fuel}\) :
-
Fuel (gas) input to CHP unit
- \({\lambda }_{i,t,s,h,\omega ,\xi }^{curtail,e}\) :
-
Electrical load curtailment at bus \(i\)
- \({\lambda }_{m,t,s,h,\omega ,\xi }^{curtail,g}\) :
-
Gas load curtailment at node \(m\)
- \({\lambda }_{p,t,s,h,\omega ,\xi }^{curtail,h}\) :
-
Heating load curtailment at node \(p\)
- \({\sigma }_{{l}^{e},t,s,h,\omega ,\xi }^{switch,e}\) :
-
Binary switching status of electrical line \({l}^{e}\) under disruption
- \({\sigma }_{{l}^{g},t,s,h,\omega ,\xi }^{switch,g}\) :
-
Binary switching status of gas pipeline \({l}^{g}\)
- \({\sigma }_{{l}^{h},t,s,h,\omega ,\xi }^{switch,h}\) :
-
Binary switching status of heating pipe \({l}^{h}\)
- \({\nu }_{i,j,t,s,h,\omega ,\xi }^{connect,e}\) :
-
Binary connectivity indicator between electrical buses \(i\) and \(j\)
- \({\widehat{p}}_{g,t,s,h,\omega ,\xi }^{gen}\) :
-
Adjusted power output of generator \(g\) under disruption
- \({\widehat{f}}_{{l}^{e},t,s,h,\omega ,\xi }^{e}\) :
-
Adjusted power flow on electrical line \({l}^{e}\) under disruption
- \({\widehat{q}}_{{l}^{g},t,s,h,\omega ,\xi }^{g}\) :
-
Adjusted gas flow in pipeline \({l}^{g}\) under disruption
- \({\widehat{h}}_{{l}^{h},t,s,h,\omega ,\xi }^{h}\) :
-
Adjusted heat flow in heating pipe \({l}^{h}\) under disruption
- \({\zeta }_{i,z,t,\omega ,\xi }^{island,e}\) :
-
Binary variable indicating if bus \(i\) belongs to island \(z\)
- \({\mu }_{z,t,\omega ,\xi }^{restore}\) :
-
Fraction of load restored in zone \(z\)
- \({\phi }_{t,\omega ,\xi }^{resiliency}\) :
-
Overall system resiliency metric
- CHP :
-
Combined heat and power
- COP :
-
Coefficient of performance
- DC :
-
Direct current (referring to the power flow model)
- P2G :
-
Power-to-gas
- RES :
-
Renewable energy sources
- RM :
-
Reserve margin
- SOC :
-
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Moshkelgosha, M., Niknam, T. & Bahmani-Firouzi, B. A tri-level stochastic framework for planning integrated electricity, gas, and heating networks with enhanced resilience and renewable integration. Sci Rep (2026). https://doi.org/10.1038/s41598-026-43763-7
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DOI: https://doi.org/10.1038/s41598-026-43763-7
