Sustainable Design of Urban Rooftop Food-Energy-Land Nexus

Summary Urban rooftop functional design offers a promising option to enable multi-function urban land-use to deliver multiple ecosystem services, e.g., food production by rooftop agriculture and energy supply by installing photovoltaic (PV) panels. To identify the best rooftop utilization strategy considering multiple decision criteria and understand the impact of rooftop solution on the design of urban energy systems, we propose a whole system modeling framework that integrates biogeochemical simulation and multi-objective energy system optimization. We apply the framework to evaluate three rooftop agriculture options, namely, basic rooftop farming, unconditioned greenhouse, and conditioned greenhouse, and one rooftop energy supply option, i.e., PV panels, for an urban energy eco-design case in Shanghai, China. Enabling rooftop agriculture options brings more flexibility to the design and operation of energy systems. PV panels provide cost-optimal solutions, whereas conditioned greenhouse potentially delivers environmentally sustainable land-use by contributing to climate regulation ecosystem services.


INTRODUCTION
Landscapes generate multiple benefits for human society and individual well-being including housing, transportation, and a wide range of ecosystem services (ES) (Mace et al., 2012; MillenniumEcosystemAssessment, 2005). These services can be broadly categorized into four categories, i.e., provisioning services, e.g., food and energy; regulating and supporting services, e.g., climate and water regulation and waste recycling; and cultural services, e.g., recreational value. Although the need to incorporate such ES into decision support at different spatial scales is increasingly recognized, their value is often overlooked in realworld land-use planning applications (Bateman et al., 2013). Over the last decade, urban ES and urban agriculture have received increasing attentions as two-thirds of the overall population is expected to be urbanized in 2050 (Cortinovis and Geneletti, 2018; Hansen et al., 2019; IEA, 2019). With rapid urbanization and the projected 50% increase of population in the twenty-first century globally (UN, 2017), food and energy demand are expected to increase 50% and 30%, respectively, between now and 2050 (EIA, 2019; Grafton et al., 2015). This will increase resource supply stress and affect land scarcity and natural ecosystems (Folberth et al., 2020). A transformation from traditional farming toward sustainable land management and urban agriculture systems is necessary where multi-functional land-use systems and urban food production sites enable sustainable food supply for the urban consumption centers (Orsini et al., 2014). However, such transformation is hindered by conflicting ES, such as climate regulation versus the food and energy provision, which compete on the limited urban land resources (Acuto et al., 2018). How this urban populace will be sustainably fed and energized is a vital focus for governments, urban planners, and academia.
Urban rooftops offer alternative resources for multi-functional land-use (e.g., housing and urban agriculture), but its potential benefits have not been well explored. Two promising solutions are proposed in this study-(1) implementing rooftop agriculture for food production and (2) installing photovoltaic (PV) panels for energy supply. Implementing rooftop agriculture has the potential to bring a range of benefits such as reducing urban heat island effect (Coccolo et al. . Each year is divided into three periods, namely, summer, winter, and transition period. Cooling is supplied in summer, heating is supplied in winter, and no heating or cooling demand is there in the transition period (see Supplemental Information Figure S5). A typical day is modeled for each period, and it is equally split into hourly intervals with varying solar conditions (see Supplemental Information Figure S4). All buildings are clustered into six zones ( Figure 2B); the urban energy system needs to be designed optimally to simultaneously fulfill the energy demands of all zones including electricity, cooling, and heating. Within each zone, one energy hub, located at the node with the largest energy demand, can be installed to supply the energy demand of that zone via an optimally designed energy network. Energy can be transmitted between energy hubs if necessary. The shortest length of the network and guaranteed connection of all buildings is achieved in each zone by the minimum spanning tree technique (Unternä hrer et al., 2017). All energy hubs are connected to the utility grids as well. Four options are available for designing rooftop utilization strategies in each zone. Figure 3 visualizes the definitions of four available rooftop options, which are closely related to the energy hub design. Installing PV panels (OPT1) can generate green electricity to reduce the reliance on grid electrical energy. In contrast, implementing rooftop agriculture options (OPT2$OPT4) is expected to bring additional economic benefits, mitigate carbon emissions, and reduce buildings' cooling and heating demands. Hence, both rooftop energy and agriculture options have the potential to contribute positively to energy hubs in terms of economic and environmental footprints.
All energy hubs are allowed to make different choices among the four available rooftop options in our modeling framework. These four available rooftop options, i.e., PV panels and rooftop tomato farming, are further defined in Table 1 and visualized in Figure 3. The proposed tool consists of two modules, i.e., biogeochemical module and energy system module. The biogeochemical module simulates the yields, inputs, and emissions of three rooftop agriculture options. This information is fed to the energy system module, by which the optimal results including the best rooftop option, energy network design, energy system configuration, and system operation strategy can be obtained. (B) The electricity, cooling, and heating demands of all buildings within one zone are served by an energy hub. The energy hub model is generalized, including six commonly used energy supply technologies, two energy storage technologies, the interactions to the grid, as well as energy network availability (Jing et al., 2019a). On the rooftop of each building, four options are available assuming the bearing capacity of rooftop is sufficient. The optimization results will determine the best choice of the rooftop option as well as the optimal energy system configurations and hourly operational strategies.

Article Biogeochemical Simulation Results
We model the tomato cultivation on rooftops considering that China is one of the main tomato producers and consumers worldwide, and that tomato is rich in nutrition and acts as a key resource of daily vitamin C intake (Wikipedia, 2020). The biomass yields and C partitioning between seed, stem, leaves, and roots obtained from the DNDC model (Li et al., 1992) simulations for one crop cycle (approximately 150 days) are given in Table 2 based on daily temperature and rainfall conditions (see Supplemental Information Figure S1).
DNDC simulated the daily carbon fluxes (see Supplemental InformationFigure S2). Gross primary production (GPP) represents the total amount of carbon fixed by photosynthesis (Wang et Table 2, DNDC projected negative NEE for one crop cycle, which indicated a net uptake of CO 2 by the plant-soil ecosystem. The NEE values vary with the options-roof farming, unconditioned greenhouse, and conditioned greenhouse can achieve À405, À974, and À1,841 kg C/ha/year, respectively. These simulation results demonstrate the beneficial effects of elevated CO 2 level in OPT4 on net carbon sequestration by plant-soil ecosystem. By incorporating the DNDC simulation results into the multi-objective optimization, the modeling framework enables urban rooftop utilization solutions to account for the biogeochemical carbon cycling.

Multi-objective Energy System Optimal Design
By integrating DNDC simulation outputs into multi-objective optimization and techno-economic parameterization, a series of different system design and rooftop utilization strategies are derived. As plotted in Figure 4, a Pareto frontier represents the trade-off between cost optimal and GHG minimization objectives, where the system design and selection of rooftop options vary significantly. Note that the cost includes the annual operation expense and the capital expenses amortized over the assumed lifetime of the project, i.e., 20 years. To further enable the decision-makers to articulate their preference of multiple decision criteria and lead to optimal solution to address the trade-offs, we apply the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method to choose one trade-off solution with maximum rationality of selection (Jing et    In the meantime, combined rooftop agriculture options (OPT2$OPT4) could significantly increase the flexibility of energy system design. As shown in Figure 4, when only OPT4 is selected, the system performance varies within a relatively narrow range (14,200-15,200310 3 USD/year for cost and 135-200 3 10 3 kg/year carbon emission) by just changing the energy system design. While different rooftop agriculture (OPT2$OPT4) and energy options (OPT1) are available, the system design flexibility significantly improved with the cost and GHG objectives varying within a wider range of 13,400-15,200310 3 USD/year and 135-445 3 10 3 kg/year, respectively.

Impact of Rooftop Options on Energy System Design
The rooftop options, as well as heating and cooling network design, for three representative solutions are visualized in Figure 5, which is consistent with the observation in Figure 4. No obvious trends can be concluded in terms of cooling and heating network design from three representative optimal solutions on the Pareto frontier. A further investigation for system implications of rooftop solutions on the energy system configuration and network design is illustrated in Figure 6.
In Figure 6, the technology sizing choices are expressed as a function of the objectives switching from costoptimal to CO 2 emission-optimal. Several observations are summarized below.
1. With the objective moving from cost-optimal to GHG-optimal, the installed capacity of combined heating and power (CHP) and boiler dropped gradually as shown in Figures 6A and 6B. This can be partially explained by the natural gas-based technologies (CHP and boiler) offering cost-competitive design options but embedding higher CO 2 emissions. To achieve lower emission designs gradually, the installed capacities of CHP and boiler have to decrease. Consequently, less residual heat is available for absorption chiller to generate cooling; a reduced installation capacity was also observed in Figure 6E. Meanwhile, PV panel (OPT1) gives its place to agriculture options (OPT2$OPT4) to fulfill the gradually higher requirement of the GHG objective.
2. As less capacity of the absorption chiller is installed, a larger capacity of the electric chiller is required to fulfill the cooling demand. With the increase in electric chiller capacity, more electricity is required. Due to the lower capacity of CHP, more electricity has to be imported from the grid with a higher import/export ratio as shown in Figure 6H.
3. The size of the cooling storage tank decreases followed by a rapid increase at the compromising point. Meanwhile, no obvious trends are found in cooling and heating network design; the cooling storage tank seems not to play signification role in the network design.
4. The variation degree for the capacity of each technology is different before and after the compromising point. From cost-optimal to trade-off solution, selecting different rooftop solutions (OPT1$OPT4) produces more impacts on system design than the energy technologies themselves. By selecting different agriculture solutions (OPT2$OPT4), CO 2 emission can be reduced efficiently. Consequently, the change of installed capacities for energy technologies is relatively moderate.
Once the optimal solution passes the compromising point, the rooftop solution is constant (all select OPT4-conditioned greenhouse), whereas capacities of energy technologies vary to further reduce CO 2 emissions.
Overall, constrained by limited land resources particularly in urban areas, rooftop offers alternative options for multi-functional land-use. As presented in this study, by integrating agriculture and energy systems with urban rooftops, the land not only delivers housing benefits for human society but also has the potential to bring multiple ES (energy and food provisioning and climate regulation ES). Different rooftop options (OPT1$OPT4) impact the ES benefits (income and climate regulation) significantly. Energy provisioning option (OPT1) brings the highest income achieving the cost-optimal rooftop land-use solution. Agriculture options (OPT2$OPT4) bring more climate regulation ES benefits than OPT1, among which OPT4 with elevated CO 2 concentration is shown as a favorable choice to achieve a GHG minimization solution.

Sensitivity Analysis
Sensitivity analyses were carried out to understand the system implications of the electricity and natural gas prices. As demonstrated in Figures 7A and 7B, the prices of the grid electricity and the natural gas significantly impact the trade-offs between cost and emissions performances. Generally, with the increase in energy price (electricity and natural gas), the whole system costs increase. Despite the variation in prices, all scenarios achieved similar GHG performances. For each scenario, moving toward the minimal cost, the emission level increases with a reduction in energy prices. This can be explained by the higher quantity of grid electricity purchased, which offers cost-efficient energy but induces higher GHG emissions compared with onsite power generation. iScience Article Figure 7C illustrates rooftop solutions along the Pareto frontier for each scenario. Although the system performances vary with energy prices, the rooftop use strategies remain relatively stable. This can explain the Pareto frontiers in Figures 7A and 7B, which are relatively evenly distributed. Overall, sensitivity analysis suggests a constant optimal solution for rooftop utilization, regardless of the energy prices variation, where PV panels (OPT1) are selected as a cost-optimal option and conditioned greenhouse (OPT4) is selected by all rooftop design to achieve minimized GHG scores.

DISCUSSION
To achieve food-energy-land nexus sustainability in an urban context, rooftop agriculture and energy systems offer promising solutions through multi-functional land-use design strategies. Four design options have been explored in the current study including solar PV panels for power generation (OPT1) and rooftop agriculture systems without and with controlled greenhouse (OPT2$OPT4). A cross-disciplinary approach has been applied to integrate biogeochemical simulation and mathematical optimization into urban energy planning decision-making framework. The developed Mixed Integer Linear Programming model enables simultaneous optimization of rooftop utilization strategies and the whole energy system design to assess the design trade-off between the minimized costs and GHGs. This essentially represents a tradeoff between provisioning and regulatory ES. Our research highlights that the PV panel (OPT1) and the rooftop greenhouse with controlled CO 2 concentration, temperature, lighting, and humidity (OPT4) offer an economically competitive and environmentally sustainable choice, respectively.  iScience Article a minor capacity variation of energy technologies, the GHG emission of the whole system can be reduced efficiently; once the rooftop solution is constant, the capacities of energy technologies need to vary significantly to further achieve lower emission design. Besides, earlier study has found that food production could be more beneficial than energy generation in Mediterranean climates through cost-benefit analysis of rooftop solutions only (Benis et al., 2018); however, the impact of rooftop solutions on the whole urban energy system has not been considered. The results could be case specific depending on various conditions, e.g., climate, type of buildings, food and energy prices, etc. All above observations, in turn, highlight the importance of developing tools that can bring land-competing systems (e.g., food and energy) and conflicting objectives into a whole system decision support framework to inform urban landscape design.
Notably, only carbon sequestration by the rooftop agricultural systems is accounted for in the model, where attributional carbon counting approach has been followed. However, the plants, e.g., tomato cultivated in rooftop agriculture systems can avoid the arable land-use elsewhere, which further leads to avoidance of GHG emissions caused by land-use. Thus, following a consequential carbon counting approach, the saving effects caused by land-use GHG avoidance could enhance the environmental competitiveness of rooftop agriculture systems. This study only accounts for the GHG emissions of the operation phase, which often dominates life cycle GHGs of an energy system (Wang et al., 2015). Future efforts are needed to integrate comprehensive life cycle assessment and multiple environmental impact indicators into the modeling framework.

CONCLUSION
Overall, the proposed modeling framework integrates for the first time biogeochemical simulation and multi-objective optimization to understand the implications of environmental variables (e.g., temperature, atmospheric CO 2 concentration) and crop-environment interaction on urban energy systems design. In this study, we first modeled different tomato cultivation options using biogeochemical simulation and developed a neighborhood-level energy system optimization model; the biogeochemical simulation results were fed into energy system model to resolve the bi-objective optimization to address the trade-offs between cost optimal and GHG emission minimization. An illustrative case study demonstrates the iScience Article applicability of the proposed decision-support tool and generates insights into the optimal design options for rooftop at a given urban neighborhood in Shanghai, China. Our research suggests that the multi-functional rooftop design from whole systems perspective enables urban food-energy-land sustainability to deliver food and energy provisioning, carbon regulation ES. The integration of agriculture options brings more flexibility to urban energy systems design when multiple conflicting objectives are considered. The PV panels provide cost-optimal rooftop solutions, whereas conditioned greenhouse potentially delivers environmentally sustainable land-use by contributing to climate regulation ES.

Limitations of the Study
This study presented a modeling framework underpinned by biogeochemical simulation and energy systems optimization to inform the rooftop utilization options. Several emerging research directions are worth further investigation efforts and highlighted below: (1) In the current study, all building rooftops were suitable for implementing rooftop farming and PV options from building structure perspective. These building rooftops were assumed to be exposed to the sun, whereas building heights and the possible shadow effects of adjacent taller buildings were not considered. However, such effects could play significant roles in some locations, and thus are worth exploring.
(2) The urban energy model we developed is based on a green-field case when designing a new building and energy system. However, an interesting research direction would be to compare new building and building retrofit. The process of retrofitting would involve a balancing of different design elements and their effects on the overall performance (e.g., energy demand, safety) of a building; thus the design criteria, design space for building retrofit, could be significantly different from rooftop design with new building. Despite the current research on a case study in the context of China, the modeling framework developed in this study could be applied to rooftop design and case studies in other urban or hinterland areas where the building patterns and underlying parameters and design criteria would vary with the region-or country-specific climate and geographical features.
(3) The optimization modeling framework developed in this study addresses the trade-off between economic and environmental objectives and considers food/energy provisioning and climate iScience Article regulation ES. However, to enable such modeling framework to provide evidences for specific rooftop design solutions, inputs and feedback from multiple decision-makers (e.g., policy makers, urban planners, households) on model feasible spaces and decision criteria are important. This can be achieved by engaging multiple model users in interactive solution-searching settings to support informed decision-making. User interaction can be explored by developing a human-in-the-loop approach in multi-level modeling research to articulate the dynamic preferences of multiple decision-makers based on their gradually built understanding of the model topology and enable the solution search to be progressively directed toward the regions of interest.
(4) The current study integrated energy and agriculture system with urban rooftops to deliver housing benefits and multiple ES including energy and food provisioning and climate regulation ES. However, other potential design criteria, e.g., building safety, stability, and wider ES, e.g., water cyclerelated ES, can be incorporated into the modeling framework proposed in this study and further developed in future research (5) The current study focuses on PV and rooftop farming; however, other potential rooftop utilization strategies including renewable energy solutions, e.g., PV-wind hybrid system and recreation park, could be explored in future research investigations. The modeling framework developed in this study could be further expanded to include wider renewable energy systems in the energy system design optimization module and simulate other plant species in the biogeochemical simulation module. Assume the grid electricity price (ele_buy) and the natural gas price (NG) vary between À40% and +40% from the baseline. For both prices, six scenarios, i.e., À40%, À20%, À10%, +10%, +20%, and +40%, are evaluated. iScience Article (6) Another interesting future research direction is to consider urban microclimate conditions and its interaction with rooftop farming and energy systems. Either PV or vegetation on rooftops could affect the urban microclimate and consequently affect the energy performance of the urban neighborhood. For example, both rooftop PV and vegetation contribute to mitigation of the urban heat island effects and further lead to lower cooling demands as well as higher power output from PV (Berardi and Graham, 2020; Dong et al., 2020). Relevant research could be further embedded in both energy supply and demand-side of the proposed modeling framework to achieve a more holistic research.

Resource Availability Lead Contact
Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Miao Guo (miao.guo@kcl.ac.uk).

Materials Availability
This study did not generate new unique reagents.

Data and Code Availability
The input data are available in Supplemental Information, and the code of energy system model associated with the article is available from the Lead Contact on reasonable request.

METHODS
All methods can be found in the accompanying Transparent Methods supplemental file.

DECLARATION OF INTERESTS
The authors declare no conflicting interests.  demand. The crop growth and development driven by air temperature, soil water, and nitrogen 24 supplement is simulated by the plant growth sub-model at a daily timestep. In the meantime, the 25 decomposition sub-model tracks turnover of soil organic matters that produce CO2 emitted from the 26 soil as well as inorganic nitrogen released from mineralization. The other three sub-models calculate 27 trace gas emissions from nitrification, denitrification, and fermentation, respectively. All six sub-28 models interact with each other to simulate the targeted ecosystem's water, C and N cycles. Overall, 29 DNDC can predict the impacts of climate change or management alternatives on the soil 30 biogeochemistry and the crop yield. 31 We use DNDC to simulate the daily tomato growth and daily net ecosystem exchange (NEE) of 32 carbon based on the detailed parameterization and simulation setup as shown in Table S1 in 33 Supplemental Information (SI). The 5-year (2011~2015) daily meteorological data (temperature, 34 precipitation) for DNDC simulations were estimated based on the data obtained from the China 35 meteorological data sharing service system (CMDC, 2018) and presented in Fig. S1, where the daily 36 max-min temperature varies within a range of -5~37 ℃ and daily rainfall ranges between 0 and 25 cm. 37 The DNDC simulated daily NEE fluxes for one crop cycle (approximately 150 days) is presented 38 in Fig. S2 for different rooftop agriculture options. The annual yield and NEE (see Table 2) along with 39 capital and operation cost breakdown are presented in Table S2. The derived income of tomato yield 40 (calculated based on Eq. 10), the NEE, the capital and operation cost, are used to parameterize the 41 energy system design optimization model. 42 Energy system design optimization 43 This study aims to optimize urban rooftop utilization to achieve multi-functional rooftop area use 44 to deliver multiple ecosystem services, i.e. food and energy provisioning and climate regulation. To 45 achieve this, we developed a multi-objective (i.e., cost and emission minimization) optimization model, 46 which is a Mixed-Integer Linear Programming model and follows a bottom-up approach that optimizes 47 the system design and operational strategy simultaneously. The optimization problem can be stated as 48 follows: 49 Given a neighborhood of 30 buildings with known rooftop areas, locations, energy demands, 50 weather conditions and available technology options for energy and food crop production, to determine 51 the rooftop utilization, energy system design and operational strategy, to achieve a series of optimal 52 system designs representing the trade-offs between two conflicting objectives i.e. the minimized 53 annualized cost and the minimized carbon emissions. The overall electricity, cooling and heating 54 demands of all buildings over an hourly interval time horizon must be fulfilled simultaneously, which 55 bound Multi-objective optimization. The ε-constraint approach is applied to solve the bi-objective (i.e., 72 cost and emissions) minimization problem. As derived in Eq. (1), the ε-constraint approach maintains 73 the f1(x) as objective function, and converts the f2(x) to a constraint by introducing a parameter of ε. 74 Hence, the bi-objective problem is converted to a typical single-objective problem (Jing et al., 2019a) , f1(x) 75 and f2(x) denote objective function of AC and ACE. 76 was assumed for both electricity and natural gas with an interval of every 10% fluctuation. Our 92 optimization results suggest that the rooftop utilisation strategy is not sensitive to the energy price 93 parameters; regardless of energy price variation, the PV panel and the conditioned greenhouse were 94 modelled as cost-effective and GHG optimal solutions respectively. 95 Model Description. In the optimization model, energy demands and prices for each year, and 96 energy conversion efficiencies (e.g. CHP efficiency) were assumed as constant. The DNDC-simulated 97 crop yields were annualized, and the constant operational costs were assumed for plantation 98 management (e.g. fertilization, irrigation). The key decision variables and parameters are defined in 99 the Table 3~Table 7 whereas the detailed parameterization is given in Table S3. Objectives. This study considers conflicting objectives, i.e., annualized cost and GHG emissions. 111 The annualized cost (AC) is calculated as Eq.

AC CAPEX FC MC GC FI
where CAPEX represents capital cost, FC denotes fuel cost, MC is maintenance cost, GC is the grid 113 cost, and FI defines food income. 114 The CAPEX includes the capital cost of all energy device, energy network, and the potential 115 construction of different rooftop agriculture options as shown in Eq. (4). Assuming the interest rate of 116 6%, the CAPEX is further annualized by multiplying a capital recovery factor (CRF) as shown in Eq. 117 (5  where E pv/CHP is electricity generated from PV panels and CHP, Q ec/ac-cool represents cooling energy 133 supply by electric chiller or absorption chiller, E in-st and Q in-st are cooling and electricity in storage, 134 respectively. 135 Grid cost (GC) can be derived from Eq. (9), depending on the electricity purchasing cost and the 136 revenue generated by the surplus electricity sold to the grid. 137 where C im and C ex are unit price of grid electricity purchasing and tariff for electricity sold back to 138 grid, respectively. E im and E ex represent the quantity of electricity purchased and sold. 139 Food income (FI) is determined by the sales income of food produced from the rooftop agriculture 140 system and the agriculture system operational cost (Eq. (10)). 141 where  where Q h-dem represents heating demand of one zone. φ agri is binary variable determining whether a 155 rooftop agriculture option (k) is chosen. Q hf(i,j) is the heating energy flow from zone i to j via network, 156 Q ac-heat is the heating consumed by absorption chiller, Q b-heat denotes the heating generated by boiler, 157 Q re-heat is the heating recovered from CHP power generation, Q hp represents the heating generated by 158 heat pump, and Lo h-pipe is the heat loss coefficient of the heating network. 159 In the cool balance, the left-hand side of Eq. (13) includes cooling demand (Q c-dem ), potential 160 cooling demand reduction (Q c-roof ) by implementing rooftop agriculture options (φ agri ), cooling charge 161 (Q cha ) into a cooling storage, and the cooling energy flowing from zone i to j (Q cf(i,j) ). The right-hand 162 side items are the cooling energy supply by electrical chillers (Q ec-cool ) and absorption chillers (Q ac-cool ), 163 the cooling energy flowing from zone j to i (Q cf(j,i) ); and the cooling energy discharged (Q disc ) from the 164 storage. A cooling loss rate (Lo c-pipe ) is applied to represent the cooling transfer loss. 165 9 c-dem agri c-roof  cha  cf( , )  , ,  ,  , , ,  , ,  , , ,  1,2,3   ac-cool  ec-cool  cf( , )  c-pipe  disc  , , ,   where the efficiency of PV panel ηpv is related to solar radiation index (SRI) in the unit of (W/m 2 ), the 178 ambient temperature (T), and the air mass (AM), SRI0 = 1000 W/m 2 , T0 = 25℃, AM0 = 1.5, P1 = 0.2820, 179 P2 = 0.3967, P3 = -0.4473, P4 = -0.093, P5 = 0.1601; Ai is the available roof area. 180 To keep the linearity of the model, the efficiency of each energy supply device is assumed to be 181 constant. Consequently, specific operation constraints are applied for the CHP avoiding low part-load 182 operations and possible efficiency drop. 183 The minimum part load constraint is set at 30% of full capacity to avoid CHP operating at a low 184 load range when the engine is on. 185      Fig. S4