Robust and optimal design of multi-energy systems with seasonal storage through uncertainty analysis

Applied Energy 238 (2019) 1192–1210

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Robust and optimal design of multi-energy systems with seasonal storage through uncertainty analysis

T

Paolo Gabriellia, Florian Fürera, Georgios Mavromatidisb,c, Marco Mazzottia,

⁎

a

Separation Processes Laboratory, ETH Zurich, Sonneggstrasse 3, 8092 Zurich, Switzerland Chair of Building Physics, ETH Zurich, Stefano-Franscini-Platz, 8093 Zurich, Switzerland c Laboratory for Urban Energy Systems, Empa Duebendorf, Ueberlandstrasse 129, Duebendorf, Switzerland b

HIGHLIGHTS

GRAPHICAL ABSTRACT

guidelines for robust and op• General timal design of MES with limited information.

resolution to describe time• Required series of input data for multi-objective design.

of most relevant input • Correlation data with robustness and optimality indexes.

and analysis of robust sce• Definition nario for multi-objective design of MES.

of uncertainty of input data on • Impact costs, emissions and robustness of MES.

ARTICLE INFO

ABSTRACT

Keywords: Multi-energy systems Time-series analysis Optimization under uncertainty Robust design Energy storage Scenarios analysis

This work proposes a framework for the robust design of multi-energy systems when limited information on the input data is available. The optimal design of a decentralized system involving renewable energy sources and energy storage technologies is considered by formulating a mixed integer linear program that determines the optimal selection, size, and operation of the system to provide energy to an end user, while minimizing its total annual costs and CO2 emissions. Different aspects related to the feasibility and the optimality resulting when operating the multi-energy system on input data different than those used for the design are studied. Input data include weather conditions, energy demands and energy prices. First, considering a single input scenario, we define the resolution required to describe the time profiles of the input data. To do so, we aggregate hourly-resolved yearly time-series through a different number of typical design days, and we define the minimum number of typical design days necessary to obtain feasible and optimal system designs when these are operated on the original input profiles. Next, the uncertainty of weather conditions and energy demands is analyzed. This is described through several scenarios, which are created by combining different climate models, greenhouse gas emissions forecasts and models of buildings. The system configurations obtained with one of these scenarios are evaluated by introducing performance indicators that quantify the robustness and the cost optimality attained when operating the selected design on all other possible scenarios. Furthermore, correlations between performance indicators, structure and size of the system, and relevant characteristics of the input scenarios are identified. Based on this analysis, a robust scenario is defined, which requires information on the average scenario only, and its performance is compared against those of the average and of the worst-case scenarios when used to design a multi-energy system. Several designs are

⁎

Corresponding author. E-mail address: [email protected] (M. Mazzotti).

https://doi.org/10.1016/j.apenergy.2019.01.064 Received 24 September 2018; Received in revised form 1 December 2018; Accepted 9 January 2019 0306-2619/ © 2019 Published by Elsevier Ltd.

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investigated, focusing on the minimization of costs and CO2 emissions, showing that the proposed robust scenario allows obtaining a robust and optimal system design through a deterministic optimization problem, hence maintaining the computation complexity at a minimum. Despite the higher robustness and cost optimality, slightly higher CO2 emissions are observed than the average and the worst-case scenarios. The proposed method is illustrated by performing the optimal design of a multi-energy system providing electricity and heat to a Swiss urban neighborhood, typical of the city of Zurich.

1. Introduction

optimality is not achieved. This gap was filled by Bertsimas and Sim, who introduced the so-called -robustness [38]; here, the degree of conservativeness is reduced by considering that only a finite number of uncertain parameters can take their worst-case values at the same time. Similarly to stochastic programming, two-stage methods have also been proposed for robust optimization based on the concept of adjustable (or adaptive) robustness [39,40]. In this case, the system operation is progressively determined when a better knowledge of the uncertain parameters is available. Downsides of robust optimization with adaptive robustness regards the fact that the complexity of the optimization problem scales with the time horizon, making it unsuitable when designing MES including long-term energy storage. Recent advances in two-stage programming are reviewed by Grossmann et al. [41]. Robust optimization tools have been applied to solve both the optimal operation [42–48] and the optimal design [49–53] of MES. Besides the aforementioned stochastic programming and robust optimization approaches, different pathways towards optimal and robust design of energy systems have been proposed [54–56]. For example, an interesting approach has been presented by Cheng et al., who coupled Monte Carlo simulations and system optimization to determine the cooling energy demand of the considered end user and the minimum-cost chiller configuration [54]. A detailed comparison between stochastic and robust optimization techniques for application to distributed urban energy systems was performed by Mavromatidis [35]. As mentioned above, drawbacks can be identified for all approaches in terms of (i) trade-off between robustness and optimality of the solution, (ii) computationally expensive algorithms, and (iii) amount of required information. Furthermore, the concept of a robust scenario implemented in robust optimization is not clearly defined for decentralized energy systems involving renewable energy sources and storage technologies, as they are characterized by highly variable and interdependent input profiles. Interesting findings were presented recently by Majewski et al., who proposed a two-stage robust approach and showed that a robust system designed on the worst-case scenario features a configuration very similar to the system designed on the nominal scenario [52]. Indeed, whereas the operation costs increased significantly for the former, the system configuration, hence the investment costs, stayed essentially unchanged. In other words, the uncertainty of the input data is more relevant in terms of the system operation than of the system design. While they considered the minimum-cost optimization of an industrial park, with no renewables integration and energy storage, we start from these considerations to investigate the design of MES including renewabled-based conversion technologies, as well as short- and longterm energy storage technologies. The robust design of such systems is significantly complicated by the large number of decision variables, by their complex interactions, and by the difficulty in defining a robust scenario accounting for all the variables of interest. For these reasons, the robust and optimal design of energy systems including different forms of energy storage at different time scales has not been presented, yet. This contribution fills this gap by providing a general tool for the robust and optimal design of MES when limited information is available and low computation complexity is requested. To do so, we investigate several aspects related to the unfeasibility and optimality of MES configurations when they are operated on input data different than those used for the design. More specifically, the work improves the current state-of-the-art by (i) defining the resolution required to describe the

Multi-energy systems (MES) are recently gaining more and more interest in the field of urban energy systems in an effort to move from a traditional, centralized energy system to a more decentralized solution, where energy is generated at the same location where it is used [1]. The prominence of MES for the future global energy system is also stressed by the Energy Technology perspective 2016 of the International Energy Agency (IEA) [2]. In general, MES combine renewable-based and conventional conversion technologies, as well as storage technologies, to reduce the economic and environmental impacts of providing energy services to end users [3]. Within the design, operation and optimization of MES, mixed integer linear programming (MILP) has been favored as optimization technique because of the combination of accurate system description and reasonable computation complexity [4]. Typically, the optimal design of MES is carried out through a deterministic MILP, implying that the input data of the optimization problem, i.e. environmental conditions, energy prices and energy demands, are known without uncertainty at the moment when the MES is designed, e.g. [5–11]. However, such data are generally affected by considerable uncertainty, making the deterministic solution possibly suboptimal or even unfeasible [12]. One common approach to deal with uncertainty is through the use of sensitivity analysis methods [13,14]. However, while these allow identifying the most influencing uncertain parameters, they do not determine the optimal system design that should be implemented. To tackle this issue, both robust optimization and stochastic programming tools have been recently applied to design multi-energy systems that guarantee the security of energy supply under uncertain operating conditions [15]. In stochastic programming the uncertain parameters are modeled through continuous or discrete probability distributions, assumed to be available. The stochastic design of distributed energy systems is often carried out through a two-stage optimization problem. First-stage decision variables include the selection and size of the available conversion and storage technologies and are fixed at the beginning. Secondstage variables refer to the scheduling and operation of the underlying technologies and are calculated later based on the uncertain parameters. Stochastic approaches have been applied to determine both the optimal operation [16–22] and the optimal design [23–33] of MES. An advantage of stochastic programming is that it provides a single optimal MES design, while calculating the optimal operation of the system for multiple scenarios, thus offering valuable information to decision makers. However, the probability distributions describing uncertain parameters are typically not known and need to be estimated, which again affects the robustness and the feasibility of the design [34]. Additionally, computational issues arise when simulating a large number of possible scenarios or when implementing complex models [35]. Robust optimization overcomes the need for probability distributions of uncertainty. Indeed, a robust solution considers every possible combination of uncertain parameters and requires information only on the range of the uncertain input data [12]. Despite its only recent application within the framework of MES, robust optimization is a relatively well known technique. More specifically, strictly robust optimization, which ensures feasibility for each scenario while minimizing the cost for the worst-case scenario, was introduced by Soyster [36] and revised by Ben-Tal and Nemirowski [37]. A drawback of strictly robust optimization is that solutions are typically over-conservative while 1193

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time profiles of a given input scenario when used for the MES design; (ii) investigating the uncertainty associated to the input scenarios and defining the concept of robust scenario when dealing with decentralized energy systems involving renewable energy sources and storage technologies; (iii) evaluating the impact of data uncertainty on costs, emissions and robustness of different MES designs obtained by minimizing costs and CO2 emissions. This paper is structured as follows. The investigated system and the corresponding deterministic optimization are described in Section 2. The analysis of the uncertainty is presented in Sections 3 and 4, while the proposed robust design methodology is discussed in Section 5. Finally, conclusions are drawn in Section 6.

where c and d represent the cost vectors associated to continuous and binary decision variables, x and y, respectively; A and B are the corresponding constraint matrices and b is the constraint known-term; Nx and Ny indicate the dimension of x and y, respectively. Within this framework, both continuous and binary variables are optimized, with the latter being introduced to model the installation, the scheduling (on/off status) and the costs of the considered technologies. In the following, the optimization problem is described in terms of input data, decision variables, constraints, and objective function; a more detailed description of such features can be found elsewhere [10]. 2.1. Input data

2. System description and optimization problem

The input data are time-dependent profiles with an hour resolution. Inputs to the optimization problem are:

The considered MES is shown in Fig. 1 and has the primary goal of supplying the electrical and thermal energy demands of a defined end user. The MES is connected to the natural gas and electricity grids and consists of a set of conversion and storage technologies including photovoltaic (PV) panels, electricity-driven heat pump (HP), natural gas boiler, natural gas proton exchange membrane fuel cell (NGPEMFC), hot water sensible thermal storage (HWTS), lithium-ion battery, and a power to hydrogen (PtH2) unit. The PtH2 system consists of a proton exchange membrane electrolyzer (PEME), a hydrogen PEMFC (H2 -PEMFC), and a hydrogen storage tank (H2 S). Note that gas and electricity can be imported/exported, but no network models are considered in this work, as the focus is on the study of the uncertainty affecting the input data of the optimization problem. A single energy hub that satisfies the electricity and heat demands is considered; therefore, the energy is converted and stored in a central node and then delivered to the buildings, where the heating network losses are taken into account as discussed earlier [10]. Moreover, no reliability issues are considered here, i.e. all technologies are assumed to be able to operate during every hour of the year. A deterministic MILP is formulated to determine the selection, size and operation of the considered conversion and storage technologies that minimize the total annual cost, or the CO2 emissions, of the system while satisfying the energy demands. It can be written in general form as

N i. The environmental conditions, namely air temperature T N and solar irradiance I , where N indicates the number of time instants in the time horizon; as data are available at every hour of the year, N = 8760. Other meteorological parameters, such as atmospheric pressure and humidity, were not considered as they are not influential to the energy demand of buildings and to the performance of conversion technologies. N ii. The electricity and heat demands, Le and Lh , respectively. iii. The prices of energy utilities, namely import prices of electricity and N and u , respectively, and export electricity gas, u e g M × N , where M is the number of available technoloprice, ve gies. While the import gas price is considered to be fixed along the year, different electricity prices are considered for different periods of the day and of the year. Also, different export electricity prices are considered for conventional and renewable-based technologies. N iv. The carbon emission rates of electricity and natural gas, e and g , respectively. While the emission rate of natural gas is assumed to be constant, the electricity emission rate changes at every hour of the year based on the average grid energy mix. v. The set of available conversion and storage technologies with the corresponding performance and cost coefficients, which are reported in Tables 1 and 2 of Ref. [10], respectively.

min (cTx + dTy)

The values of input data considered for the presented case study are described in more details in the Appendix A. The uncertainty related to the aforementioned input data is discussed in Sections 3 and 4.

x, y

s. t. Ax + By = b x

0

Nx ,

y

(1)

{0, 1} Ny

Fig. 1. Schematic representation of the investigated multi-energy system. Illustrative end user from Ref. [35]. 1194

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2.2. Decision variables

where i and µi represent the variable and fixed cost coefficients for the i-th technology, respectively; yi indicates whether the i-th technology is installed or not; ai is the annuity factor of the i-th technology, which is used to annualize the capital cost. All cost coefficients, which are either constant or size-dependent, are reported in Table 2 of Ref. [10]. The annual operation cost is calculated based on the amount of imported and exported electricity and gas:

The values of the following decision variables are the outputs of the optimization problem: i. The selection of available technologies, y {0, 1} M . M ii. The size of the installed technologies, S . iii. The scheduling (on/off status) of the conversion technologies, x {0, 1} MC × N , where MC is the number of available conversion technologies. iv. The input and output power of the available technologies, F and M×N P , respectively. MS × N v. The energy stored in the storage units, E , where MS is the number of available storage technologies. N , respectively, vi. The imported electricity and gas, Ue and Ug M × N and the exported electricity Ve .

N

Jo = j {e, g } t = 1

V j , i, t

j, t i=1

t

(6)

The uncertainty investigated in this work pertains to the environmental conditions (ambient temperature and solar irradiance) and to the energy demands (electricity and heat) of the end user under consideration. The uncertainty associated to the energy prices and to the emission footprint of the energy grids depends on political and economic considerations, whose analysis is beyond the scope of this work; they are typically tackled with sensitivity analysis methods [60], though a robust optimization study has also been proposed [61]. Furthermore, whereas energy prices have an effect on the optimality of MES, they do not affect their feasibility. The analysis of the uncertainty follows a recent work [35] and is based on the creation of several possible scenarios. Future climate conditions are predicted by combining two categories of climate models with different spatial resolutions, namely General Circulation Models (GCM), characterized by a spatial resolution of about 200 × 200 km, and Regional Climate Models (RCM), featuring a spatial resolution of about 20 × 20 km. Several combinations of global and regional climate models have been developed by different institutions and are openly available [62,63]. The resulting climate models consider different scenarios of greenhouse gas emissions to determine possible evolutions of average temperature and climate. Such emissions scenarios are compiled by the Intergovernmental Panel

2.4. Objective function The objective function of the optimization problem is the total annual cost of the system, J, given by the sum of the capital, operation and maintenance contributions. The annual capital cost is expressed as M

( i Si + µi yi ) ai

Uj, t

3. Definition of uncertain scenarios

(2)

(3)

i=1

M j, t

j {e , g } t = 1

where i indicates the i-th technology.

Jc =

of the annual

(5)

N

e=

M

Lj , t = 0

(4)

In addition to the total annual cost, the system is evaluated in terms of environmental performance, namely annual CO2 emissions. This translates into a multi-objective optimization problem, which is solved through the -constraint method [59]. First, a minimum-cost and a minimum-emissions optimizations are solved to determine the achievable range of CO2 emissions. Then, minimum-cost optimizations are solved where the annual CO2 emissions are constrained below given values spanning this range. The annual CO2 emissions are calculated based on the amount of imported and exported electricity and gas:

i. Performance of conversion and storage technologies. The performance of the fuel cell, electrolyzer and storage systems are modeled as described earlier [57,58] (following methodology TM I in [58] for the fuel cell and the electrolyzer), whereas the performance of PV panels, HP and boiler are modeled as described in [10]. For all conversion technologies, approximate affine or piecewise affine correlations are implemented which take into account partial-load performance, conversion dynamics, as well as minimum-power requirements. ii. MES energy balances. These state that for all the considered energy carriers, namely electricity (e), heat (h), natural gas (g) and hydrogen (H2 ), the sum of imported and generated power must equal the sum of exported and used power. Therefore, for all energy carriers j {e, h, g, H2} and for all time instants t {1, N } :

Fj, i, t ) + Uj, t

i Jc, i i=1

The constraints of the optimization problem can be grouped into two categories:

Vj, i, t

t

M

Jm =

2.3. Constraints

(Pj, i, t

vj, i, t Vj, i, t i=1

The annual maintenance cost is given as a fraction capital cost:

The decision variables (i) and (ii) define the design of the system, while the decision variables (iii)-(vi) characterize its operation.

i=1

M

uj, t Uj, t

Table 1 Climate and demand scenarios characterizing the uncertainty of the input data. Climate models [62,63] CNRM-CERFACS-CNRM-CM5-CCLM4 ICHEC-EC-EARTH-CCLM4 ICHEC-EC-EARTH-HIRHAM5 ICHEC-EC-EARTH-RACMO22E IPSL-IPSL-CM5A-MR-WRF331F MPI-M-MPI-ESM-LR-CCLM4

Emission scenarios [64]

RCP4.5 RCP8.5

Time span

2021–2040

Building parameters [35] 6 combinations of: occupant density, equipment capacity, material properties, hot water demand, ventilation rates, thermostat settings

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on Climate Change (IPCC) and are typically referred to as Representative Concentration Pathways (RCPs); four RCPs are provided, namely RCP2.6, RCP4.5, RCP6, and RCP8.5, which reflect the variation of radiative forcing values in the year 2100 relative to pre-industrial values [64]. In this work, we consider six pairs of GCM-RCM applied to two emission scenarios, namely RCP4.5 and RCP8.5. The former represents a moderate-intervention scenario, with a predicted increase in ambient temperature of 1.7–3.2 °C for the late twenty-first century (2081–2100 average) relative to the 1850–1900 average, whereas the latter is the business-as-usual scenario, with a predicted increase in ambient temperature of 3.2–5.4 °C [65]. Such emission scenarios are chosen because considered representative of realistic possible future scenarios and because of abundance of data. The results of such climate models, in terms of ambient temperature and solar irradiance time profiles, are available until year 2100 within the framework of the CORDEX project [62]; we considered a period of 20 years, from 2021 to 2040. Such profiles are available with a daily temporal resolution, and are transformed into hourly-resolved time-series using the “morphing” technique presented elsewhere [66]. Starting from these climate scenarios, possible evolutions of electricity and heat demands are simulated by using the building performance simulation tool EnergyPlus [67]. In addition to the environmental conditions, such demands depend on building-related uncertain parameters, such as (i) occupant density, (ii) lighting/equipment capacity, (iii) material properties, (iv) hot water daily demand, (v) ventilation rates and infiltrations, (vi) thermostat settings. Each of these parameters is described through independent probability distributions (see Table 5.5 of Ref. [35], as well as the Appendix B below). Six different configurations of such parameters are considered for each climate scenario.

All the resulting scenarios are summarized in Table 1: 6 climate models are applied to 2 IPCC emission scenarios and the results considered for 20 years; this results in 240 environmental scenarios, i.e. 240 hourly-resolved yearly profiles of ambient temperature and solar irradiance. Then, energy demands profiles are generated by considering 6 different configurations of building-related parameters for each climate scenario, thus resulting in 1440 demand scenarios. In the following, we assume that the defined pool of scenarios accurately describes the uncertainty of the input data for the investigated system. Using the available dataset an “average” scenario is created to represent the type of information that is usually available to MES designers. Such an average scenario is created by first defining a Typical Meteorological Year (TMY) following the method presented in [68]. In brief, the method identifies typical months based on the aforementioned 240 environmental scenarios and combines them to create the TMY. Then, using the TMY all the buildings of the considered end user are simulated with EnergyPlus with deterministic, nominal profiles for the occupant patterns, equipment, temperature and ventilation settings. The environmental conditions of the TMY and the resulting energy demand patterns are henceforth referred to as the average scenario, which represents the reference scenario to design the MES and the starting point to build the robust scenario developed in this paper. While the proposed procedure is independent of the considered application, we illustrate the analysis by referring to a typical urban neighborhood based in Zurich, Switzerland. The neighborhood is described in detail elsewhere [35] and consists of 24 buildings including single-family houses, multi-family houses, office and commercial buildings. The years of construction of the buildings are chosen to be representative of the age of the building stock in Zurich. All resulting scenarios, as well as the average scenario (black line), are shown in

Fig. 2. Scenarios characterizing the uncertainty of the considered input data: (a) mean monthly temperature, (b) global monthly horizontal radiation, (c) monthly thermal demand, (d) hourly thermal demand along the first two months of the year. The average scenario based on the TMY and is indicated by the black line. The TMY is created using the method presented elsewhere [68] with modified weight factors for the weather variables to match the scope of this paper: 0.5 for the global solar radiation, 0.25 for the mean dry bulb temperature and 0.125 for the min and the max dry bulb temperatures. 1196

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Table 2 Strategy adopted to determine the number of typical days required to represent a single input scenario. I II III IV V

Simplification of the original time profiles through a given number, D, of typical design days Design and operation of a reference MES by using the original time profiles (CO2 emissions cap for intermediate values of CO2 emissions) Design of MES by using simplified time profiles obtained with different numbers of typical design days (CO2 emissions cap for intermediate values of CO2 emissions) Operation of the MES designed by using the simplified time profiles on the original time profiles (fixed selection and size of the available technologies, no CO2 emissions cap) Calculation of the discrepancy in objective function between the reference MES and the simplified MES when operated on the original time profiles

Steps II–V are repeated for different values of CO2 emissions. Different MES designs are obtained by minimizing costs while imposing different maximum values of CO2 emissions. Such emissions caps are considered only at design phase.

Fig. 2, which reports (a) the mean monthly temperature, (b) the global monthly radiation, (c) the monthly thermal demand, and (d) the hourly thermal demand along the first two months of the year.

time-series are typically described through aggregated periods, e.g. typical design days [69–71]. However, an excessive simplification can lead to suboptimal or unfeasible MES configurations, leading to questioning the robustness of the system design. Here, we aim at determining the resolution required to represent a single input scenario through typical days. While this feature was already studied earlier [10] for the minimum-cost design of a system including power to hydrogen (through the comparison between methods M1 and M2 for modeling the time horizon), here we study the behavior of a generic system for different levels of CO2 emissions (by using method M1 introduced in [10]). To do so, we consider the average scenario defined in the previous section and we apply the procedure reported in Table 2. First, MES are designed by considering the original input profiles as well as simplified input profiles, which are obtained by describing the original time-series through different numbers, D, of typical design days. Then, the obtained MES designs are

4. Uncertainty analysis In the following, different aspects related to the feasibility and optimality of a MES design are investigated through the analysis of the uncertainty affecting the input data. 4.1. Required resolution for a single input scenario Even when adopting a deterministic approach, by considering an average or a specific scenario to design the MES, the remarkable complexity of the optimization problem often requires simplified profiles of the input data. More specifically, the original hourly-resolved

Fig. 3. Resolution required when simplifying single input scenario. (a)–(c) Comparison between original hourly-resolved yearly thermal demand of the average scenario and its description through 3, 12, and 48 typical design days. (d) Discrepancy in the objective function between the reference MES and the simplified MES designs. Three values of CO2 emissions considered: minimum-cost design (blue circles), characterized by a reference cost Jref = 0.07 €/kWh (eref = 76.2 gCO2 /kWh); minimum-emissions design (green diamonds), characterized by a reference emissions eref = 29.6 gCO2 /kWh; −50% CO2 emissions design (orange squares), characterized by a reference cost Jref = 0.09 €/kWh and a maximum CO2 emissions of emax = 52.9 gCO2 /kWh. 1197

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operated on the original input profiles and the discrepancy in objective function is evaluated. All the optimizations are solved by using the commercial software Gurobi [72], set up to have a relative MIP gap of 1%. The set of design days is identified by using the MATLAB-embedded clustering algorithm k-means [73,74]. Based on environmental conditions, energy prices and demand profiles, the most representative set of D design days is determined, and every day of the year is assigned to a specific design day, characterized by its own typical hour resolution. Fig. 3-a,b,c compare the original thermal demand of the average scenario and the simplified thermal demands obtained when considering 3, 12, and 48 typical design days, respectively. Fig. 3-d reports

the discrepancy in objective function between the reference MES, which is designed and operated on the original input profiles, and different simplified MES, which are designed by using different numbers of design days and operated on the original input profiles, as a function of the number of design days. The objective function normalized over the total annual demand is reported. Three optimizations are considered, which provide (i) the minimum-cost design, with the objective function being the total annual cost, J; (ii) the minimum-emissions design, with the objective function being the annual CO2 emissions, e; (iii) the design characterized by a CO2 emissions reduction of 50% (imposed through a constraint on the maximum value of annual CO2 emissions at design phase), with the objective function being the total annual cost. The

Fig. 4. Top: schematic representation of the investigated MES; set of available technologies (left) and selected configurations (right) for the three considered optimizations. Bottom: size of the installed technologies for the minimum-cost design (blue circles), the minimum-emissions design (green diamonds) and the −50% emissions design (orange squares). The PtH2 system is represented by the H2 storage. Reference designs represented by the dashed lines. 1198

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value of CO2 emissions reduction is defined by referring to the range identified by the minimum-cost and the minimum-emissions optimizations, i.e. 0% reduction for the minimum-cost design and 100% reduction for the minimum-emissions design. The emissions constraint is imposed only at design phase, and not at operation phase. Concerning the minimum-cost optimization (blue circles), for D below a certain threshold, here D = 18, the designs obtained through the simplified input profiles are unfeasible when operated on the original input profiles, meaning that they cannot satisfy the energy demand of the enduser. For D above such a threshold, feasible but suboptimal designs are obtained, meaning that a cost higher than the reference MES design is obtained; the suboptimality decreases for increasing D, due to the better description of the original input profiles (in line with what found in [10]). Going from the minimum-cost to the minimum-emissions optimization (from the blue circles to the green diamonds), a lower number of typical design days is required to obtain a feasible design. However, a greater suboptimality, i.e. a greater discrepancy in objective function, is observed for the same number of typical design days. The higher feasibility obtained at lower CO2 emissions depends on the structure of the MES design, i.e. technology selection and size. Indeed, going from the minimum-cost to the minimum-emissions optimization a wider portfolio of technologies is selected and bigger sizes are installed, which results in the possibility of satisfying the energy demands in several ways. The selected MES configurations for the considered values of CO2 emissions are shown in the upper part of Fig. 4. It is worth noting that (i) heat pump and hot water thermal storage are the only technologies that are always installed; (ii) the boiler is not selected in the minimum-emissions design, which exhibits a higher degree of electrification to satisfy the thermal demand (electrification is particularly convenient in Switzerland due to the low carbon footprint of the grid electricity); (iii) PV panels are not selected in the minimum-cost design because they are too expensive; (iv) battery and hydrogen storage are installed only in the minimum-emissions design, as the possibility of reducing CO2 emissions by storing the surplus in renewable generation comes at a very high cost with respect to the grid electricity. In fact, battery and hydrogen storage play a role only when the entire surplus of renewable generation must be exploited, both at short- and long-term scale, to minimize the carbon emissions [10]. The sizes of the installed technologies for the considered values of CO2 emissions are shown in the lower part of Fig. 4. The size of the short-term (daily, weekly) storage units, i.e. hot water thermal storage and battery, is systematically underestimated when considered simplified input profiles. On the contrary, the size of longterm (seasonal) storage units, i.e. hydrogen storage, is systematically overestimated when considering simplified input profiles, in line with previous results [10]. Note that thanks to the big size of the installed storage technologies, the minimum-emission design is able to exploit the low carbon footprint of the electricity grid in summer, hence reducing the CO2 emissions during the winter season. The higher suboptimality observed at lower CO2 emissions depends on the connection between system operation and CO2 emissions. More specifically, the wider portfolio and the bigger size of the installed technologies obtained when reducing CO2 emissions result in higher investment and overall costs, but allow reducing the energy imported from the grid, which is responsible for the CO2 emissions as well as for

the operation costs. To fully exploit this operation advantage, a more detailed description of the short- and long-term dynamics of the input profiles is necessary, which requires a large number of typical design days. In the following, the original hourly-resolved profiles are considered by using the methodology M2 introduced earlier [10], which does not require using typical design days to describe the input data. Note that the information on the hourly-resolved profiles of an average scenario is typically available at design phase. 4.2. Robustness and optimality indicators So far, we have considered a single set of input data and we have defined the resolution of the corresponding time-series required to obtain a feasible and optimal MES design. Now, we aim at investigating the uncertainty associated to the input data and at defining strategies to obtain a robust and optimal MES design when considering all possible input scenarios. To do so, robustness and optimality indicators are needed to evaluate the performance of a MES that is designed on sce, and including all nario a and operated on scenario b, with a, b the 1440 scenarios discussed in Section 3. The evaluation procedure is described in Table 3. The robustness, R, is the ability of the system to fulfill the energy demands. Here it is defined as the fraction of the energy demand that can be actually delivered by a system designed on a when operated on b, averaged for all b . Since the system is connected to the electricity grid, which is assumed to be able to provide electricity at every hour of the year, in the following robustness is based on the thermal demand only. Thus, the robustness of a MES designed on scenario a is given by

Ra =

1 Y

Rab = b

1 Y

b

Hab Hb

(7)

where Y = 1440 is the number of considered scenarios, Hab the total annual thermal demand satisfied by a system designed on scenario a when operated on scenario b, and Hb the total annual thermal demand of scenario b. If the thermal demand can be satisfied at every hour of the year, for all scenarios b , with a MES designed on scenario a, then the robustness Ra = 1; any violation of the thermal demand penalizes R. To account for the possibility of violating the energy demand, the thermal energy balance given by Eq. (2) is modified for all t {1, N } as M

(P h, i, t

Vh, i, t

Fh, i, t ) + Uh, t

Lh, t =

h, t

(8)

i=1

where h, t is non-negative and indicates the violation of thermal demand at the time instant t. A fictitious arbitrarily high cost u is associated to the energy violation to prevent the optimizer from identifying this as the optimal solution. Thus, the operation cost given by Eq. (4) is modified as N

Jo =

M

uj, t Uj, t j {e, g } t = 1

N

vj, i, t Vj, i, t

t+

i=1

u

h, t

t

t=1

(9)

5

Here, u is selected to be equal to 10 €/kWh, thus implying that the optimal MES violates the thermal demand only when it cannot do

Table 3 Strategy adopted to determine a robust and optimal MES design when considering all possible scenarios. I II III

Design and operation of MES on all scenarios in (MILP described in Section 2) Operation of the MES designed on scenario a on all other scenarios b (fixed selection and size of the available technologies, no CO2 emissions cap)

Evaluation of the MES design in terms of costs, J (see Eqs. (3)–(5)), emissions, e (see Eqs. (6)), robustness, R (see Eqs. (7)), and optimality, O (see Eqs. (10))

Steps II–III are repeated for different values of CO2 emissions. Different MES designs are obtained by imposing different maximum values of CO2 emissions. Such emissions caps are considered only at design phase, while only costs are minimized during the operation phase

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otherwise. Note that the concept of robustness, which quantifies how much a design is unfeasible, is not introduced in Section 4.1 because of the nature of the analysis. Whereas in Section 4.1 the MES design was unfeasible due to an excessive simplification of known input profiles, here the MES design is unfeasible due to the lack of knowledge about possible future scenarios. The cost optimality, O, measures how much the cost of a system deviates from the ideal case where such a system is designed and operated on the same scenario. Here it is defined as the ratio of the cost of a MES designed and operated on scenario b to the cost of a MES designed on scenario a and operated on scenario b, averaged for all b . Thus, the cost optimality of a MES designed on scenario a is given by

Oa =

1 Y

Oab = b

1 Y

b

Jbb Jab

defined as above). All MES designs are obtained by using the average . scenario as design scenario a and are operated on all scenarios b At operation phase the structure of the MES is fixed (technology size and selection) and no CO2 emissions constraint is imposed. The clouds of transparent symbols refer to single operation scenarios, i.e. 1440 , whereas the solid symbols refer to values of Rab and Oab for all b the average on all available scenarios, i.e. Ra and Oa . One can note that going from the minimum-cost to the minimum-emissions design, higher values of robustness (always higher than 99.6%) and lower values of cost optimality are observed. Differently from the others, the minimumemissions design always results in very high robustness, but presents a much higher variability of cost optimality compared to the other cases, which show an increasing optimality range when increasing the robustness. Similar to Section 4.1, the trends of robustness and cost optimality depend on the different MES configurations and on the different trade-offs between design and operation costs obtained for the different CO2 emissions levels (see Fig. 4). More specifically, the wider portfolio and the bigger sizes of installed technologies obtained when reducing the CO2 emissions result in higher robustness, but also in higher investment and overall costs, which translate in a wider range of cost optimality. As an example, take two MES designs performed on scenario a, which has low energy demands, and scenario b, which has high energy demands. On the one hand, when minimizing the costs the MES design obtained with a would have a smaller boiler, hence a slightly smaller investment cost, than the MES design obtained with b. When operating both designs on b, the operation costs would be predominant in the calculation of the overall costs, and the difference in optimality would be negligible. On the other hand, when minimizing the emissions the MES design obtained with a would have smaller PV, battery, PtH2, hence a significantly smaller investment cost, than the MES design obtained with b. When operating both designs on b, the operation costs would play a minor role in the calculation of the overall costs, and the difference in optimality would be remarkable. The performance of the considered MES design in terms of CO2 emissions and costs is shown in Fig. 5-b. Going from the minimum-cost to the minimum-emissions design, a smaller costs variability and a greater emissions variability are observed. On the one hand, reducing the emissions of 50% from the minimum-cost value results in a slight increase of the system cost, a smaller emissions variability, and about

(10)

where Jab is the total annual cost of a system designed on scenario a when operated on scenario b, and Jbb is the total annual cost of a system designed and operated on the same scenario b. It is worth noting that cost optimality can be greater than 1. A cost optimality Oab > 1 means that a system designed on scenario a is more cost effective than a system designed on scenario b when both are operated on scenario b; this can happen because of a lower energy demand of scenario a, and implies necessarily a penalization of the robustness. On the contrary, a cost optimality Oab < 1 implies that the scenario a leads to an overconservative design when operated on scenario b, with no implications on robustness. It is worth noting that the evaluation of a MES design does not necessarily require to operate it on all possible scenarios. Indeed, taking as correct the values of robustness and cost optimality obtained when considering the whole set of scenarios, an accurate estimation of both indicators can be obtained by using only a subset of scenarios. Whereas all available scenarios are considered in the following to avoid introducing errors in the evaluation process, the impact of the number of operation scenarios on the accuracy of the calculation of R and O is discussed in the Appendix C. Fig. 5-a shows the robustness and cost optimality of four MES designs obtained for different values of CO2 emissions, namely the minimum-cost, the −50% emissions, the −90% emissions, and the minimum-emissions designs (with the emissions reduction values

Fig. 5. Performance indicators of four MES designs obtained by using the average scenario as design scenario and optimized to obtain the minimum-cost design (blue circles), the minimum-emissions design (green diamonds), the −50% emissions design (orange squares), and the −90% emissions design (purple triangles). The designs are evaluated on the R O plane (a) and on the e J plane (b). The clouds of transparent points refer to single operation scenarios, i.e. 1440 values of R ab and Oab for all b , whereas the solid points refer to the average on all available scenarios, i.e. R a and Oa . 1200

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the same cost variability. On the other hand, a further 50% emissions reduction to obtain the minimum-emissions design results in a remarkable increase of the system cost, a smaller emissions variability, and a much higher cost variability. Again, this depends on the selected MES configurations and on the trade-off just discussed between design and operation costs.

Indeed, similar to the MES design obtained with the average scenario and shown in Fig. 4, all the considered scenarios install a wider portfolio of technologies (including PtH2), as well as bigger conversion and storage units when minimizing the CO2 emissions. Referring to the top of Fig. 4, the MES designs for a 90% CO2 emissions reduction look like those for a 50% CO2 emissions reduction with also battery installed. The effect of robustness on the MES structure is presented in Fig. 7, which shows the size of the installed technologies for the considered values of CO2 emissions. For each technology, only the relevant values of CO2 emissions are reported, i.e. those where some correlation between size and robustness is observed. In general, higher sizes of the installed units allow increasing robustness. More specifically, (a) a higher size of the heat pump allows increasing robustness for all levels of CO2 emissions but for the minimum-cost designs, where no correlation is observed; (b) a higher size of the boiler increases the robustness of the minimum-cost designs, whereas no correlation is observed for the −50% CO2 designs, and no boiler is installed for lower values of CO2 emissions; (c) PV panels are not relevant from a robustness perspective, with no correlation observed for the −50% CO2 designs and a constant size for lower values of CO2 emissions; (d) a higher size of the sensible thermal storage allows increasing robustness for all levels of CO2 emissions but for the minimum-cost designs, where no correlation is observed; (e) a slight correlation is observed between the size of the battery and the robustness of the −90% CO2 emissions designs, whereas no battery is selected for higher values of CO2 emissions and a constant size is installed for the minimum-emissions designs; (f) the robustness of the minimum-emissions design is increased for bigger sizes of the PEM electrolyzer (PEME), which allows generating H2 when needed within the framework of the PtH2 system. The exponential law defined by Eq. (11) can be used to calculate the robustness for a given value of h, which only requires knowing the average scenario. While this is generally available at design phase, its representativeness is affected by the uncertainty associated to the definition of the future scenarios. While it is assumed here that the available scenarios represent well the uncertainty of the input data, the impact of an uncertain average scenario is discussed in the Appendix D. Concerning the cost optimality, a correlation is observed with the total annual demand of the year used to design the MES. This is shown in Fig. 8-a, which reports the cost optimality of all scenarios a a as a function of l, defined as the total annual demand of the design scenario normalized over the total annual demand of the average scenario. Similar to h, the calculation of the value of l only requires knowing the average scenario. An optimal value of l, leading to a maximum cost optimality for all values of CO2 emissions, is observed. The presence of a maximum depends on the trade-off between installation and operation costs. On the one hand, low values of l result in smaller sizes of the

4.3. Robustness and optimality correlations Whereas the robustness and the cost optimality of the average scenario were shown in the previous section, any scenario within the available pull can be used at design phase. The goal of this section is to define correlations between the features of a given input scenario used at design phase and the resulting performance indicators. Several features describing the input scenarios are considered. They are summarized in Table 4 and include the (i) annual energy demands and solar irradiance, (ii) maximum hourly, daily, weekly energy demands, ambient temperature and solar irradiance, (iii) maximum daily, weekly variability (from 10 to 90% of maximum value) of energy demands and ambient temperature, (iv) thermal-to-electrical demand ratio. The procedure defined in Table 3 is applied to a subset a of 100 with a uniform scenarios, randomly chosen out of the entire pool probability distribution. Each of the 100 scenarios is used to design a MES that is then operated on all 1440 available scenarios and evaluated in terms of robustness and cost optimality. For each scenario a, the same four levels of CO2 emissions considered above are studied. Partial least square regression is adopted to identify which scenario features explain best the behavior of the resulting values of R and O [75]; the Matlab-embedded algorithm is used [76]. Robustness proves to correlate well with the maximum daily thermal demand of the scenario used to design the MES. Interestingly, the maximum hourly thermal demand has no influence on the design robustness. This is because short peaks in thermal demand do not require further conversion or storage capacity, whereas a high demand over periods longer than a few hours require additional conversion or storage capacity that must be installed at design phase. Because of similar considerations, the maximum value of thermal demand over periods longer than a day also shows little influence on the system robustness. Also the time location of the maximum daily thermal demand does not play a role in the definition of robustness, provided that it occurs during months with high thermal demands, i.e. winter season in this case.This suggests that time shifts of the energy demand do not affect the system robustness, provided that the daily and seasonal dynamics are unchanged. Fig. 6-a shows the robustness of the MES designs as a function of h, defined as the the maximum daily thermal demand of the scenario used to design the MES normalized over the annual thermal demand of the average scenario. For all emissions values, R increases with h according to the following exponential correlation:

R=1

e

h

Table 4 Investigated input data and corresponding characteristics.

(11)

Energy demands

where and are constant coefficients, varying for different CO2 emissions values. Whereas a similar behavior is observed until a CO2 emissions reduction of 90% with respect to the minimum-cost value, a sharp variation is observed for the minimum-emission designs. The figure shows an exponential fitting characterized by = 55.6 and = 9.74 , which is obtained by considering three emissions values, but neglecting the minimum-emission case; the fitting coefficients are determined using the Matlab-embedded function for least square nonlinear fitting [77]. Despite its strong impact on robustness, h affects optimality only slightly, as shown in Fig. 6-b. This means that no significant trade-off between R and O is necessary when setting a robust target up to, for example, 99.9%. In fact, the minimum-emission case (not considered in the definition of the exponential correlation) is not particularly interesting from a robustness perspective, as very high values are always obtained.

Annual electricity demand Annual thermal demand Annual total demand Maximum hourly thermal demand Maximum 1–7-day thermal demand Time of the maximum daily thermal demand Maximum 1–7-day thermal demand variation (10–90%) Thermal-to-electrical demand ratio Environmental conditions Maximum ambient temperature Maximum 1–7-day ambient temperature variation (10–90%) Annual average ambient temperature Maximum solar irradiance Maximum 1–7-day solar irradiance Annual solar irradiance

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Fig. 6. Correlation of robustness (a) and cost optimality (b) to the normalized maximum daily thermal demand for 100 scenarios, randomly selected with a uniform distribution among the entire pool. Four different MES designs are considered for each scenario, namely the minimum-cost design (blue circles), the minimumemissions design (green diamonds), the −50% emissions design (orange boxes), and the −90% emissions design (purple triangles). The emissions caps are imposed only at design phase. (a) The exponential correlation approximating R as a function of h is reported (black line); R2 = 0.9304.

installed technologies, and thus in lower installation costs. On the other hand, smaller sizes can lead to a less efficient system operation, and thus higher operation costs. The optimal value of l identifies the point where the increase in operation costs dominates over the decrease in

investment costs. Despite its impact on cost optimality, l affects robustness only slightly, as shown in Fig. 8-b. This means that no significant trade-off between O and R is necessary when working at the optimal value of l,

Fig. 7. Size of the installed technologies as function of robustness for the minimum-cost design (blue circles), the −50% CO2 emissions design (orange squares), −90% CO2 emissions design (purple triangles), and minimum-emissions design (green diamonds). For each technology, only the relevant CO2 emissions levels are reported, i.e. those showing some correlation with robustness. The PtH2 system is represented by the PEM electrolyzer (PEME), as the sizes of fuel cell and H2 storage do not show clear correlations. 1202

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Fig. 8. Correlation of cost optimality (a) and robustness (b) to the normalized total annual demand for 100 scenarios, randomly selected with a uniform distribution among the entire pool. Four different MES designs are considered for each scenario, namely the minimum-cost design (blue circles), the minimum-emissions design (green diamonds), the −50% emissions design (orange boxes), and the −90% emissions design (purple triangles). The emissions caps are imposed only at design phase. (a) An optimal value of l leading to the maximum cost optimality is observed, which is similar for all values of CO2 emissions.

which in this case is about 0.88 for all values of CO2 emissions. This corresponds to reducing the total annual demand of about two-thirds of the variability observed across all input scenarios. The effect of optimality on the MES structure is illustrated in Fig. 9,

which shows the size of the installed technologies for the considered values of CO2 emissions. Similar to robustness, only the relevant values of CO2 emissions are reported for each technology, i.e. those where some correlation between size and optimality is observed. In general,

Fig. 9. Size of the installed technologies as function of optimality for the minimum-cost design (blue circles), the −50% CO2 emissions design (orange squares), −90% CO2 emissions design (purple triangles), and minimum-emissions design (green diamonds). For each technology, only the relevant CO2 emissions levels are reported, i.e. those showing some correlation with optimality. The PtH2 system is represented by the PEM fuel cell (PEMFC) and the PEME, as the size of the H2 storage remains constant with optimality. 1203

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larger sizes of the installed units result in a lower cost optimality. More specifically, (a) a higher size of the heat pump results in lower optimality for all values of CO2 emissions but for the minimum-emissions designs, where no correlation is observed; (b) the size of the boiler does not really affect optimality, although a slight correlation is observed for the minimum-cost designs; (c) increasing the size of the PV panels leads to a lower cost optimality for the −50% emissions designs, whereas a constant size is installed for lower values of CO2 emissions; (d), (f) higher sizes of the PtH2 units, namely the PEM fuel cell (PEMFC) and the PEME result in a lower cost optimality for the minimum-emissions designs, which is the only case where the PtH2 is installed; (e) a larger size of the battery leads to a decrease in cost optimality for the −90% emissions designs, whereas no battery is selected for higher values of CO2 emissions and a constant size is installed for the minimum-emissions design. Being a fairly cheap technology, the considered hot water thermal storage is not relevant from an optimality perspective and it is not shown in the figure.

and can be lowered to increase cost optimality, (ii) the maximum daily thermal demand defines robustness and does not really impact cost optimality. Thus, starting from the average profiles, the electrical and thermal energy demands are lowered to obtain the optimal value of total annual energy demand, l (see Fig. 8). Then, the maximum daily thermal demand, h, is increased according to the target value of robustness (see Fig. 6). All other input data remain unchanged with respect to the average scenario. Note that the definition of the robust scenario requires the knowledge of the average scenario only. The MES designs obtained with the average, worst-case, and robust scenarios are illustrated in Fig. 10, which shows the size of the installed technologies as a function of the reduction in CO2 emissions for the four emissions levels considered above. The reduction in CO2 emissions goes from 0% for the minimum-cost designs to 100% for the minimumemissions designs. The structures of the MES designs remain unchanged with respect to those obtained for the average scenario and shown in Fig. 4. In general, bigger sizes are installed going from the minimumcost to the minimum-emissions designs and going from the average to the worst-case scenarios. Concerning the designs obtained with the robust scenario, a trade-off is observed between the aspects discussed with reference to Figs. 7 and 9 to increase robustness and optimality, respectively. More specifically, the following is observed. (i) The size of all technologies increases going from minimum costs to minimum emissions, except that of the boiler which decreases (Fig. 10-b). (ii) The worst-case MES designs feature the biggest sizes for all values of CO2 emissions reduction and for all technologies, except for the thermal storage, which is most exploited by the robust MES designs (Fig. 10-d). (iii) Until a CO2 emissions reduction of 50%, the robust MES design limits the size of the heat pump with respect to the average scenario to increase optimality (Fig. 10-a), and it increases the size of the boiler and of the thermal storage to increase robustness. (iv) For values of CO2 emissions reduction above 50%, the boiler disappears from the MES structure and the robust MES designs rely more heavily on the heat pump and on the thermal storage to increase robustness. In this case, the optimality is preserved by reducing the size of expensive technologies, namely the battery (Fig. 10-e) and the PtH2 system (PEMFC, Fig. 10-f). The latter is represented here by the PEM fuel cell, as the sizes of PEM electrolyzer and H2 storage do not change significantly. Note that decreasing too much the size of the electrolyzer may lead to a decrease in optimality as shown in Fig. 9-f. (v) The installed area of PV panels is similar for all scenarios. (vi) The PtH2 system is installed only in the minimum-emissions designs, since the possibility of storing H2 allows storing energy for long time periods with negligible energy losses, but comes at very high costs and low round-trip efficiencies, as discussed elsewhere [10]. Such MES designs result in the performance shown in Fig. 11 in terms of (a) robustness and optimality, and (b) CO2 emissions and costs. The average scenario is the same reported in Fig. 5. Fig. 11-a shows that using the worst-case scenario to design the MES results in R = 1 for all levels of CO2 emissions. However, it leads to values of O lower than those obtained by using the average scenario; while such a decrement in cost optimality is fairly limited for the minimum-cost design (in line with what observed in Ref. [52]), it increases when decreasing the CO2 emissions, due to the higher investment costs that characterize the more sustainable MES designs. Although the investment costs are predominant in the calculation of the total annual cost, the bigger units installed in the worst-case MES designs allow reducing the operation costs, resulting in lower CO2 emissions than the average scenario, as shown in Fig. 11-b. On the other hand, Fig. 11-a shows that using the robust scenario introduced in this work allows increasing both robustness and optimality with respect to the average scenario. A robustness in the proximity of 99.9% is obtained, consistently with the set target (h = 1.1%), with a cost optimality that is still higher than that of the average scenario. Symmetrical to the worst-case scenario, the increment in cost optimality is fairly limited for the minimum-cost design, but it increases when

5. Robust MES design The uncertainty analysis performed in the previous section shows, on the one hand, that a MES designed on the average scenario results in good values of robustness and cost optimality, provided that the input time profiles are described with a sufficient resolution to represent the daily and seasonal variability of the input data. On the other hand, it is found that robustness and optimality can be further improved by considering a design scenario characterized by high values of the maximum daily thermal demand, h, and by low values of the total annual energy demand, l. Here, we aim at (i) defining a design scenario leading to high robustness and optimality at the same time, denoted hereafter as robust scenario, (ii) comparing the performance of MES designs obtained with the robust scenario against those obtained with the average and the worst-case scenarios. Whereas the average scenario is built by considering a typical meteorological year and a nominal building configuration starting from all scenarios in (see Section 3), a worst-case scenario is built as described in Table 5: the average energy demands are increased so as their annual values match the maximum value among all scenarios, whereas the average ambient temperature and solar irradiance are lowered such that their annual values match the minimum value among all scenarios. Although the worst-case scenario ensures that the energy demands are satisfied at every hour of the year, it generally leads to overconservative MES designs characterized by high costs. Therefore, we define a robust scenario that ensures that the energy demands are satisfied “almost” at every hour of the year, i.e. fixing a target value of robustness, and maintains low costs at the same time. The definition of the robust scenario is summarized in Table 5 and is based on the considerations that (i) the total annual energy demands do not really affect robustness Table 5 Definition of worst-case and robust scenarios for design of MES. Worst-case scenario Average electrical and thermal demands multiplied by factors such that their annual values match the maximum value among all scenarios Average ambient temperature and solar irradiance multiplied by factors such that their annual values match the minimum value among all scenarios Robust scenario I II III

Starting from the average profiles, the electrical and thermal energy demands are lowered to obtain the optimal value of total annual energy demand, l (here l = 0.88 , see Fig. 8-a) The maximum daily thermal demand, h, is increased according to the target value of robustness (here h = 1.1% to obtain R = 0.999 following the exponential correlation given by Eq. (11), see Fig. 6-a) The average profiles are used for ambient temperature and solar irradiance

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Fig. 10. Size of the installed technologies as function of the reduction of CO2 emissions for four MES designs obtained by using the average (red), worst-case (black) and robust (blue) scenarios as design scenarios and optimized to obtain the minimum-cost design (circles, 0% emissions reduction), the −50% emissions design (squares), the −90% emissions design (triangles), and the minimum-emissions design (diamonds, 100% emissions reduction). The PtH2 system is represented by the PEMFC, as the size of the PEME follows the same behavior and the size of the H2 storage is the same for all scenarios.

decreasing the CO2 emissions. Although the investment costs are predominant in the calculation of the total annual cost, the smaller sizes of battery and PtH2 installed in the robust MES designs result in higher operation costs, hence in higher CO2 emissions than the average scenario, as shown in Fig. 11-b. Such an increase in CO2 emissions becomes less pronounced when going from the minimum-cost to the minimum-

emissions designs, where the bigger heat pump and storage installations allow limiting the energy import. Overall, the approach presented in this section allows the robust and cost optimal design of an energy system via the creation of a robust scenario that is used for the MES design. Additionally, unlike stochastic programming models, which use multiple scenarios to capture the

Fig. 11. Performance indicators of four MES designs obtained by using the average (red), worst-case (black) and robust (blue) scenarios as design scenarios and optimized to obtain the minimum-cost design (circles), the −50% emissions design (squares), the −90% emissions design (triangles), and the minimum-emissions design (diamonds). The designs are reported on the (a) robustness-optimality plane, R-O, and (b) CO2 emissions-cost plane, e-J. 1205

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Single scenario without uncertainty Definition of the required resolution to describe the time-series of one specific scenario (see Table 2) Uncertainty characterization Creation of climate and demand scenarios characterizing the uncertainty of the input data of interest (see Table 1) Uncertainty analysis • Definition of metrics to quantify the performance of an energy system when operated on scenarios different from that used for the design (see Table 3) • Prediction of system performance based on average input data, i.e. information available at design phase (see Figs. 6, 8) Worst-case scenario Creation of a worst-case scenario through uncertainty characterization (see Table 5): targeting robustness

Robust scenario Creation of a robust scenario through uncertainty analysis (see Table 5): targeting robustness and optimality

Robust and optimal design of energy systems Optimal design of the energy system of interest through comparative assessment of robust, worst-case and average scenarios in terms of costs, CO2 emissions and fulfillment of demand Fig. 12. Schematic of the proposed methodology.

nature of uncertainty and have very high computational requirements, the presented approach uses only a single scenario for the design, keeping the computation requirements at similar levels as with traditional deterministic design models. Moreover, the development of the robust scenario requires information that is typically available to MES designers, i.e. nominal profiles of input data spanning the duration of a year. Therefore, the proposed approach can be readily adopted by MES designers to increase the robustness of their designs. Finally, a comment is worth about the applicability and the portability of this study. We believe that our contribution is both portable to case-studies different from that presented here and applicable to real applications. On the one hand, we tackled the optimal design of MES by accounting for the whole complexity of the optimization problem, and by investigating several aspects related to the robustness and to the optimality of the system design. Therefore, the general methodology, which is summarized in Fig. 12, can be applied to different case-studies, independently of the geographic location and type of neighborhood, by deriving robust scenarios suited for the sets of input data of interest. On the other hand, the findings of the analysis can be applied with no modifications when designing real-world energy systems that are characterized by time-series similar to those considered in this paper, i.e. similar daily and seasonal dynamics of the energy demands. In fact, this has been done in Ref. [78], where the proposed robust scenario has been successfully applied to optimally design the MES for a different urban neighborhood.

urban neighborhood, typical of the city of Zurich, is considered as case study to illustrate the proposed methodology. First, starting from a single input scenario known with certainty, we define the resolution required to describe the time profiles of the input data. To do so, we approximate hourly-resolved yearly time-series through a different number of typical design days, and we define the minimum number of design days necessary to obtain feasible and optimal MES designs. Findings show that the resolution of the input profiles is relevant at design phase, and that an excessive simplification of such profiles through a low number of design days leads to MES configurations that are unfeasible or suboptimal when operated on the original input scenarios. In fact, the impact of the resolution of the input time-series increases when going from minimum-cost to minimum-emissions optimizations, due to the correlation between system operation and CO2 emissions and the consequent necessity of better describing the daily, weekly and seasonal data variability when reducing the environmental impact. Next, the uncertainty of weather conditions and energy demands is analyzed. This is described by several scenarios, which are created by combining different climate models, greenhouse gas emission forecasts and building models. The MES configurations obtained with one of these scenarios are evaluated by introducing performance indicators that quantify the robustness and the cost optimality when operating the underlying design on all other possible scenarios. Results show that the impact of the uncertainty of the input data varies across the considered values of CO2 emissions, with higher values of robustness and lower values of cost optimality observed when going from minimum-costs to minimum-emissions designs. Furthermore, correlations between performance indicators, structure and size of the MES, and relevant characteristics of the input scenarios are identified. They show that (i) robustness increases with the maximum daily thermal demand of the design scenario according to an exponential correlation, and increases for higher sizes of boiler, heat pump and thermal storage; (ii) cost optimality presents a maximum when studied as a function of the total annual demand of the design scenario and increases for lower sizes of heat pump, PV, battery and power to hydrogen. Based on this analysis, a robust scenario is defined and its performance compared against those of the average and of the worst-case scenarios when used to design MES. Though the robust scenario allows increasing robustness while reducing the overall costs of the system with respect to the average scenario, it results in higher CO2 emissions

6. Summary of main results and concluding remarks 6.1. Summary This work proposes a framework for the robust design of multi-energy systems (MES) when limited information on the input data is available. The problem of the optimal design of a decentralized MES involving renewable energy sources and energy storage technologies is considered by formulating a mixed integer linear program that determines the optimal selection, size, and operation of the system to provide energy to an end-user, while minimizing its total annual costs and CO2 emissions. Different aspects related to the feasibility and the optimality resulting when operating a given MES configuration on input data different than those used for the design are studied. A Swiss 1206

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due to a less efficient operation, caused by the smaller sizes of the installed units. On the contrary, while the worst-case scenario results in bigger sizes than the other scenarios, leading to higher installation and overall costs, it allows reducing the CO2 emissions of the system thanks to a more efficient operation. The costs increments are more pronounced for minimum-emissions optimizations, and greater emissions savings are observed for minimum-cost optimizations. The definition of the robust scenario requires only information on the average scenario and it allows obtaining a robust and optimal MES design through a deterministic optimization problem, hence maintaining the computation complexity at a minimum.

izing energy generation, storage and consumption, (ii) considering the uncertainty of future input data, and (iii) investigating different objective functions, such as total annual costs and CO2 emissions. In this work, all these aspects are taken into account. The thorough analysis of the resulting optimization problem allows identifying the most relevant quantities in terms of system design, thus defining general guidelines for the optimal design of MES when limited information is available. Furthermore, it allows obtaining a deeper understanding of the system, defining the rationale behind such design guidelines and explaining why different system structures emerge when designing MES with different objective functions.

6.2. Concluding remarks

Acknowledgements

The optimal design of MES with renewable energy generation and energy storage is a complex optimization problem due to the amount of input data, their interactions, and their uncertainty. This complexity can be taken into account by (i) formulating a detailed optimization problem, which is able to describe the different dynamics character-

This work was supported by the Swiss National Science Foundation (SNF) under the National Research Program Energy Turnaround (NRP70), Grant No. 407040-153890 (IMES project). The authors thank the whole IMES project team, for the useful discussions, data and information exchange during the project execution.

Appendix A. Values of the input data for the considered case study Whereas the method presented in the paper is general, it is applied here to design the multi-energy systems of a urban neighborhood grounded within the Swiss framework. As mentioned in Section 2, the import gas price is considered to be fixed along the year at 0.06 €/kWh [79], while different electricity prices are considered for different periods of the day and of the year based on the data for Zurich area [80]. A high tariff (HT) is considered for weekdays from 7:00 to 20:00 and Saturday from 7:00 to 13:00, while a low tariff (LT) is considered for the remaining times. The former is 0.15 €/kWh in winter (Oct-Mar) and 0.13 €/kWh in summer (Apr-Sep), whereas the latter is 0.09 €/kWh in winter and 0.07 €/kWh in summer. Note that end-user prices are not as volatile as the electricity spot market price and are not expected to rise dramatically in the foreseeable future [81,82]. The price of exported electricity depends on the different technologies and is set equal to 0.08 €/kWh for the conventional conversion technologies and to 0.1 €/ kWh for the PV panels (based on the Swiss Feed-in-Tariff [83]). Similarly, while a constant grid emission rate of 0.237 tonCO2 /MWh is considered for natural gas, values of the electricity emission rate are derived based on [84–86] and vary between 0.02 to 0.24 tonCO2 /MWh. It is worth noting that in Switzerland the electricity grid intensity is relatively low due to the major fraction of hydro and nuclear power, which add up to about 95% of the installed Swiss capacity [87,88]. Appendix B. Characterization of the uncertainty of building-related parameters The probability distributions characterizing the building-related parameters affecting the energy demands are summarized in Table 6. Table 6 Uncertainty distributions for parameters related to building energy demand. Table taken from [35]. Nominal values for material properties and infiltration rates are taken from [89,90]. SIA 2024 refers to the norm [91]. Parameter

Probability distributions

Distribution parameters

Material properties Infiltration (ACH) Occupancy density [m2/pers] Lighting capacity [W/m2] Equipment capacity [W/m2] Hot water demand [W/m2] Ventilation rates [m3/(h pers)] Thermostat settings [°C]

Normal, (µ, Normal, (µ, Triangular, ( Triangular, ( Triangular, ( Triangular, ( Normal, (µ, Normal, (µ,

µ= nominal value, µ= nominal value, = SIA2024 min, = SIA2024 min, = SIA2024 min, = SIA2024 min, µ= nominal value, µ= nominal value,

) ) , , , , ) )

, , , ,

) ) ) )

= 0.10µ = 0.10µ = SIA2024 = SIA2024 = SIA2024 = SIA2024 = 0.10µ

nom, nom, nom, nom,

= SIA2024 = SIA2024 = SIA2024 = SIA2024

max max max max

= 1 °C

Appendix C. Investigation of number of scenarios required to evaluate a MES design The impact of the number of operation scenarios on the accuracy of the calculation of robustness and optimality is shown in Fig. 13. On the lefthand side, the estimation of robustness and optimality is shown when testing the average-scenario-design on 1000 subsets of 100 scenarios each (blue dots), as well as on the entire pool of scenarios (red dot). The scenarios determining the subsets are selected randomly from the whole set by using a uniform distribution. The figure highlights the deviations between the minimum and maximum values of robustness and optimality obtained with a subset of 100 scenarios. On the right-hand side such deviations in robustness (red squares) and optimality (blue circles) are presented as functions of the number of operation years forming the subset. Findings show that by using more than 300 uniformly selected operation years (about 20% of the total number of scenarios), robustness and optimality are estimated with a deviation lower than 1% with respect to the whole set. Within this work, the MIP gap of the optimization problem, identifying the intrinsic accuracy of the optimization results, is also set to 1% (see Section 2), making 300 years a suitable number of operating scenarios to evaluate the system design. Moreover, one can note that the robustness deviation is smaller than the cost optimality, independently of the size of the subsets.

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Fig. 13. Left: estimations of robust and optimality when testing the average-scenario-design on 1000 subsets of 100 scenarios each (blue dots) and on the entire pool of scenarios (red dot); the deviations between the minimum and maximum values of robustness and optimality obtained with a subset of 100 scenarios are highlighted. Right: deviation between the minimum and maximum values of optimality (blue circles) and robustness (red squares) obtained with subsets of 100, 200, 300, 500 and 1000 operation scenarios, selected from the entire pool with a uniform distribution.

Appendix D. Impact of the uncertainty associated to the creation of scenarios The uncertainty associated to the definition of the future scenarios affects the quality of the available average scenario and therefore the calculation of the normalized quantities h and l. The accuracy of the calculation of robustness depends on the accuracy of the value of h, the maximum daily thermal demand normalized over the thermal demand of the average scenario. In its turn, the representativeness of the value of h depends on the capacity of the average scenario to represent the pool of all future scenarios. The gray shaded box in Fig. 14 shows the range of robustness that can be spanned when targeting a value of 99.9%, corresponding to h = 1.1%, assuming a thermal demand of the average scenario varying between ± 10 % of the reference value (as a consequence of a smaller capacity of the average scenario to represent the pool of future scenarios). In this case, the box goes from the value of R corresponding to h = 0.99% to that corresponding to h = 1.21%, i.e. from 99.75% to 99.95%. It is worth noting that due to the shape of the R-h correlation the higher is the target value of robustness (high h), the lower is the impact of the scenarios uncertainty. This is illustrated by the blue shaded box, obtained when targeting a robustness of 99%, which is significantly wider than the 99.9% box. Similar considerations hold for the cost optimality, related to the value of l, the total annual demand normalized over the total annual demand of the average scenario. In this case, the variability of optimality depends on the value of CO2 emissions. Due to the higher variability, a given variation of l results in a greater variation of O when going from the minimum-cost to the minimum-emission designs.

Fig. 14. (a) Correlation of robustness, R, with normalized maximum daily thermal demand for 100 scenarios randomly selected, with a uniform distribution, among the entire pool. The gray and blue shaded regions indicate the robustness variability obtained when targeting a robustness of 99.9% and 99%, respectively, and assuming an average thermal demand varying between ± 10 % of the reference value. (b) Correlation of optimality, O, with normalized total annual demand for 100 scenarios randomly selected, with a uniform distribution, among the entire pool. The colored shaded regions indicate the optimality variability obtained when targeting the maximum optimality for all CO2 emissions levels, and assuming an average total annual demand varying between ± 7 % of the reference value. 1208

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