ECOS

PROCEEDINGS OF ECOS 2018 - THE 31ST INTERNATIONAL CONFERENCE ON EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS JUNE 17-22, 2018, GUIMARÃES, PORTUGAL

Robust Optimization of District Heating Networks Structure and Dimension combining Metamodels and Multi-Objective Optimization Marius Reicha, Mario Adama and Jonas Gottschalda a

ZIES, Centre of Innovative Energy Systems, University of Applied Sciences Düsseldorf, Muensterstr.156, 40476 Duesseldorf, Germany, [email protected]

Abstract: In order to support the layout of district heating networks (DHN) or, more generally, energy supply systems (ESS) of any type, the complex behavior needs to be modelled as a mathematical representation. For modelling, various simulation environments are accessible with the easiest being the commonly used spreadsheet calculation. Whatever simulation environment the user decides for, the boundary conditions such as the global radiation or the gas price are often defined as deterministic . The accordingly defined optimal target value thus only applies to the provided boundary conditions and might vary strongly if its quantity changes. Due to that, robust optimization (RO) aims at providing a solution insensitive of uncertain boundary conditions while still being as close to the optimal target value as possible. An approach is presented that combines Taguchis idea of robust process optimization with experimental designs for computer experiments and metamodeling. The approach is applied to a DHN simulation model, which considers four fossil fuel and regenerative heating appliances. Boundary conditions are selected using results of a sensitivity analysis, including times series with an hourly resolution and a one-year time span. For RO, a two-step methodology is utilized: First, control and noise parameters are combined and used to approximate an ecological and economical target with Artificial Neural Networks (ANN). Second, control and noise parameters are separated and the ANNs are used to approximate the mean and standard deviation. Using both approximation models, the robustness can be evaluated with high efficiency as demonstrated on the DHN simulation model. The results do not only show the importance of robust optimization with regards to the layout of ESS but also give a sense of the high capability of the approach. Appropriate methods to further reduce computational efforts are named, which might also lead to applications in non-scientific environments.

Keywords: District Heating Network, Robust Optimization, Metamodel, Taguchi, Multi-Objective Optimization.

1. Introduction Deterministic simulations do not take into account any uncertainties. If such a model is optimized, influential boundary conditions that are challenging to define accurately in the planning process, might lead to a significant deterioration of the optimum when shifted from their nominal value. Robust optimization therefore endeavors to achieve an optimal setting of the factor levels, which achieves the smallest possible variance of the target variable at an optimal mean value. For optimizing energy supply systems, the negative impact of uncertain boundary conditions is well known for some time and is already being incorporated in current optimization models. For example, the authors in [1] include unsafe load forecasts in the planning of a robust operating mode. Mixed integer linear programming (MILP) is used, which is known to produce problems with high resolution time series due to the exponential growth in computational efforts [2]. In addition, many other applications are available which use MILPs to optimize different energy systems with a wide variety of target values and noise factors [3-6]. An outlier is described in [7] where a robust optimization of the life-cycle costs as well as greenhouse gas emissions is carried out that uses the non-renewable energy consumption as a boundary condition. The authors use a sampling-based approach to analyze the effect of noise on the deterministic optimum. Further methods of robust optimization are fuzzy programming [8] and min-max regret analysis [9-10]. All methods have in common that either a lot

of expensive evaluations are required and/or that the system has to be simplified to a considerable extent, coupled with restrictions for further analysis. The robust optimization according to Taguchi [11-13] is one of the most noted methods to consider uncertainties. The Taguchi method combines two highly fractional designs (crossed Arrays) with Signal-to-Noise ratios and analysis of means in order to divide control parameters into parameters that influence the expected value and the dispersion. With these divided control parameters, a robust optimum can be achieved. Although Taguchi's work is highly praised as a milestone in robust optimization, there are some limitations and problems. This includes the fact that, due to the highly fractional designs, the robust optimum is not very accurate when confronted with nonlinear problems [14]. Moreover, a high number of simulations is required due to the use of crossed arrays. The Taguchi method also leaves little space for robust multi-objective optimization. This paper presents a two-step method for a robust optimization of a local heating network structure and dimension or sizing, respectively, in which, among other boundary conditions, hourly resolved time series can be considered as uncertain. The two-step method is based on regression analysis using artificial neural networks and is outlined in Chapter 2. The goal of the first step is an approximation of the economic and ecological target value as a function of control and noise parameters, where in the second step, the mean value and the standard deviation of the two targets are to be estimated as a function of the control parameters. The approximation of mean value and standard deviation allows both, a separate, unweighted and a combined, weighted optimization. The latter enables robust optimization of economic and ecological targets, respectively. Chapter 3 describes the simulation model of the local heating network, including its uncertain boundary conditions. The application of the methodology is presented in Chapter 4, where Chapter 5 summarizes the results of the optimization and Chapter 6 provides an outlook with innovative approaches of a continuative reduction of the computational efforts. This paper focuses on the explanation of the methodology and reduces the physical analysis to a minimum.

2. Two-Step Methodology for Efficiently Evaluating the Robust Optimum The methodology presented here allows an intuitive and practical optimization of simulation models while taking into account uncertainties of the boundary conditions. The optimization process can be subdivided into two steps: The aim of the first step (Step I) is to approximate the target variables 𝑦𝑘 (with 𝑘 = 1. . 𝑛𝑘 and 𝑛𝑘 = number of targets) as a function of the control parameters 𝑥𝑖 (with 𝑖 = 1. . 𝑛𝑖 and 𝑛𝑖 = number of control parameters) as well as the noise parameters 𝑧𝑗 (with 𝑗 = 1. . 𝑛𝑗 and 𝑛𝑗 = number of noise parameters). Then, by further processing this approximation, the second step (Step II) allows to approximate the expected value as well as the dispersion of the control parameters caused by the variation of noise parameter. For doing this, various factor level combinations of the control parameters, which are defined in an inner array (IA), need to be varied by a fixed number of factor level combinations of the noise parameters, which are defined in an outer array (OA). This procedure results in a characteristic distribution of the target variables for each factor level combination of the inner array and can be described using the mean value and its standard deviation. As mentioned in the introduction (Chapter 1), these two characteristics can be used to describe the robustness of the solution and are approximated analogously to Step I. Moreover, weighting the two approximated characteristics ultimately yields a multi-objective robust optimum. Step I: Approximation of Targets, combining Design and Noise Parameters As mentioned in the introduction, the aim of Step I is to approximate the targets by combining both, control and noise parameters, resulting in a combined array (CA). The approach of the first step is visualized in Fig. 1 and is explained as follows. 2

I.1 Initialization of Design Space and Data Sampling: The CA is used to vary design and noise parameters and to measure as well as to analyze the variation in responses. Since the simulation of the system, at the given control and noise parameters, is highly time consuming, one obviously should perform only as many simulation as required in order to obtain a predefined approximation accuracy. To assure that the full design space, defined by the lower and upper bounds of the control and noise parameters, is covered, a (fractional) factorial design is used as an initial design. I.2 Metamodeling of Targets: Subsequent to a simulation of each factor level combination of the initial design, the results are used to create a model of high generalizability for each target. In this paper, feed forward artificial neural networks (ANN) are created to fulfil this requirement [15]. Due to the unknown optimal topology, a multitude of ANNs with random topology is created, trained and assessed accordingly in terms of generalizability for each target. I.3 Iteratively Appending new Data: If the mean generalizability of the ANNs do not satisfy a given threshold (which will in all likelihood be the case after the initial CA), new evenly distributed factor level combinations, e.g. using Latin Hypercube Sampling, are added to the CA and simulated accordingly. The new dataset is than again used to train ANNs. I.4 Determining Deterministic Optimum: Utilizing the trained ANNs according to 1.3, the deterministic optimum can be determined which is important for the comparison. This is done by optimizing (here: minimizing) the approximation 𝑦̂𝑘 in terms of the design parameters 𝑥𝑖 in the feasible region while keeping the noise parameters 𝑧𝑗 to their nominal values. Thus the optimization can be formulated as in (1): min 𝑦̂𝑘 (𝑥𝑖 , 𝑧𝑗,𝑛𝑜𝑚 ) , 𝑠. 𝑡. 𝑔(𝑥𝑖 , 𝑧𝑗,𝑛𝑜𝑚 ), 𝑥𝑖

(1)

The models achieved in Step I not only generate a deterministic optimum. The robust optimum can also be found using the adapted Taguchi methodology as described in Step II. Step II: Approximation of Robustness Targets - Robust Optimization of Control Parameters With the approximation of the target values as a function of the control and noise parameters, created in Step I, it is possible to approximate the mean and the standard deviation very efficiently. These robustness characteristics can again be approximated by ANNs, thus allowing robust optimization. The approach of the second step is visualized in Fig. 2 and is explained in more detail below. II.1 Definition of Inner and Outer Array for Variation of Control and Noise Parameters: Instead of using a CA like in Step I, control and noise parameters are separated into two independent designs. The design for controlling the noise parameters is called outer array (OA) and is fixed in size, whereas the design for the control parameters is called inner array (IA). The IA is, analog to Step I, initially initialized with a solid fundament of factor level combinations and then expanded iteratively to meet the ANNs fit at a certain generalizability threshold. Every combination of IA and OA is eventually approximated, thus leading to a so called response field (RF).

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Fig. 1. Flow chart to approximate the targets as a function of control and noise parameters (StepI).

II.2 Calculation of Mean and Standard Deviation: The approximated RF depicts the variation of the targets for each combination of control parameters due to the variation of noise parameters. It can accordingly be used to approximate the mean and standard deviation of each factor level combination of control parameters. Mean and standard deviation for a certain 𝑥 are calculated as defined in (2) and (3): 𝑛𝑗

∑𝑗=1 𝑦̂𝑘,𝑗 (𝑥,𝑧𝑗 )

𝑦̂𝑘,𝑚𝑒𝑎𝑛 =

𝑛𝑗

,

𝑛𝑗

(2)

∑𝑗=1(𝑦̂𝑘,𝑗 (𝑥,𝑧𝑗 )−𝑦̂𝑘,𝑚𝑒𝑎𝑛 )

𝑦̂𝑘,𝑠𝑡𝑑 = √

𝑛𝑗 −1

2

,

(3)

The iterative training of ANNs, coupled with the additional evenly distributed data, is the same as in Step I - except for the increased number of targets to approximate. 4

II.3 Determining Robust Optimum using weighted Sum and Multi-objective Optimization: Since there are two characteristics (mean and standard deviation) to characterize 𝑘 target responses to the variation of 𝑧𝑗 , a multitude of so called Pareto optimal solutions exists. One way to find an optimum is to assign weights to both characteristics. To emphasis both characteristics with the same weight, the normalized sum can be used as shown by (4). A normalization of both characteristics takes place by using the maximum measured values. Optimizing (4) ensures a Pareto optimal solution that gives a good trade-off between both characteristics. Using unbiased multi-objective optimization, one can utilize (4) to find the robust Pareto optimal solutions for different targets. 𝑦̂

min 𝑦̂𝑘,𝑅𝑂𝐵 (𝑦̂𝑘,𝑚𝑒𝑎𝑛 , 𝑦̂𝑘,𝑠𝑡𝑑 ) = 𝑦̂ 𝑘,𝑚𝑒𝑎𝑛 𝑥𝑖

(𝑥𝑖 )

𝑘,𝑚𝑒𝑎𝑛,𝑚𝑎𝑥

𝑦̂

+ 𝑦̂ 𝑘,𝑠𝑡𝑑

(𝑥𝑖 )

𝑘,𝑠𝑡𝑑,𝑚𝑎𝑥

, 𝑠. 𝑡. 𝑔(𝑥𝑖 , 𝑧𝑗 ),

(4)

Fig. 2. Flow chart of the procedure to determine the robust optimum combining ANNs and Taguchis inner and outer arrays (Step II). 5

3. Simulation Model with uncertain Boundary Conditions This chapter describes the simulation model of the district heating system as well as the uncertain boundary conditions utilized for robust optimization. A basic scheme of the system can be found in Fig. 3.

3.1. District Heating System Simulation Model The district heating system is modelled according to the energy balance method in the form of a spreadsheet, i.e. the heat demand (𝐸ℎ𝑑 ) is covered by the generation park at any time in the simulation year. The electricity demand in the district is not taken into account, electricity from the combined heat and power plant (CHP) is completely exported to the public grid. A model is created for each source, i.e. fossil-fired CHP module, ground source heat pump (GSHP), solar collectors (STC), solar energy storage (SES) and natural gas boiler (NGB). For this purpose, a large number of generalized device-specific characteristic data (including investment cost functions and efficiency curves) as well as annual curve lines (e.g. outdoor temperature, global radiation and heat demand) are stored in hourly resolution for the location Duesseldorf, Germany. A fixed power-on sequence is used to control the individual heat generators in order to cover the heat demand: Given the case that solar heat is present in the storage tank or the collector field can provide power, this is primarily used to cover the heat demand. If the heat requirement exceeds the available solar heat, the CHP module goes into operation first, followed by the electric heat pump if the output of the former is not sufficient. Consequently, the NGB goes into operation at peak load times and is set to a constant thermal power of 1 MW to ensure that the heat load is covered for every possible configuration of heat generator or supplier during optimization, respectively. Thus, possible constraints of heat load coverage in the hereafter optimization can be neglected. The target values are defined as the CO2-emissions (𝑦𝐶𝑂2) and the heat price (𝑦𝐻𝑃 ) of the energy supply system. The CO2-emissions of the system are calculated as stated in (5):

𝑦𝐶𝑂2 = 𝑓𝑔𝑎𝑠 ∙ (𝐸𝐶𝐻𝑃,𝑔𝑎𝑠 + 𝐸𝑁𝐺𝐵,𝑔𝑎𝑠 ) + 𝑓𝑒𝑙,𝑑𝑒𝑚𝑎𝑛𝑑 ∙ 𝐸𝐺𝑆𝐻𝑃,𝑒𝑙 − 𝑓𝑒𝑙,𝑓𝑒𝑒𝑑 ∙ 𝐸𝐶𝐻𝑃,𝑒𝑙 , (5) Where 𝐸 is the annual energy requirement for the indexed supplier, 𝑓𝑒𝑙,𝑑𝑒𝑚𝑎𝑛𝑑 is the CO2-emission factor for electricity that is demanded and 𝑓𝑒𝑙,𝑓𝑒𝑒𝑑 for electricity that is fed into the grid.

Fig. 3. Scheme of the district heating system used for robust optimization. 6

The calculation of the heat price includes capital costs (e.g. investment costs), maintenance and operating costs (e.g., energy costs, subsidies, taxes, etc.), as seen in (6):

𝑦𝐻𝑃 =

∑𝑛(𝐶𝑖𝑛𝑣,𝑛 ∙𝐴𝑛 +𝐶𝑚𝑎𝑖𝑛𝑡,𝑛 )+∑𝑖(𝐸𝑖 ∙𝑃𝑏𝑢𝑦,𝑖 )−𝐸𝑒𝑙,𝐶𝐻𝑃 ∙𝑃𝑠𝑒𝑙𝑙,𝑒𝑙 −𝐹𝐾𝑊𝐾𝐺 𝐸ℎ𝑑

,

(6)

The index 𝑛 indicates the supplier, 𝑖 the utilized energy source. 𝐶𝑖𝑛𝑣,𝑛 are the investment costs, including government subsidies if applicable, 𝐶𝑚𝑎𝑖𝑛𝑡,𝑛 represents the maintenance and repair costs. 𝑃𝑏𝑢𝑦,𝑖 and 𝑃𝑠𝑒𝑙𝑙,𝑒𝑙 are the prices of energy and natural gas in the purchase or sale. 𝐹𝐾𝑊𝐾𝐺 combines the subsidy components from the German legislation and 𝐴𝑛 is the specific annuity.

3.2. Uncertain Boundary Conditions Potentially, all boundary conditions (or parameters) can be considered as uncertain, which means that they cannot be precisely determined over the course of time of the actual operation and are producing a deviation of the target variable from the actual deterministic design value. Therefore, a common procedure consists of selecting boundary conditions that are potentially uncertain. In a first sensitivity analysis, these boundary conditions are examined with regards to their mean effect on the target values. For reducing computational efforts, only the boundary conditions which trigger the highest deviation are to be further analyzed in a robust optimization. Selected boundary conditions for the sensitivity analysis are: - the heat load, which is assumed to be uncertain due to the possible change in quantity of consumers as well as improving or degrading heat demand due to change of individual technology performance (e.g. thermal insulation), - the electricity refund of the electricity generated by the CHP, which is mostly influenced by politics and thus can be assumed to be of a high uncertain nature, - the gas price, since its traded on the stock market and thus can be considered uncertain - and the CO2-emission factor of the public grid due to its dependency on politics (e.g. due to deactivation of nuclear power plants, financial aid for renewable supplier) as well as improvements of technologies. Alongside those rather political or environmental boundary conditions, those concerning the performance of the energy system are also to be included. This is (among other things) due to the deviation of manufacturer's data from the real measured data as well as due to the uncertainty of efficiency curves used for the simulation. The considered boundary conditions are namely the COP of the GSHP, the nominal efficiencies of the CHP and NGB as well as the thermal loss of the SES. Although for most of the uncertain boundary conditions the nominal values as well as their expected deviation have been extracted and derived from various publications, more detailed work has to be put into such an analysis. Table 1 summarizes the noise parameters as well as their variation, which is applied as a global factor for each time series, used for the analysis of means (ANOM). The ANOM is performed by keeping the selected noise parameter at its low and high level, respectively, while varying the other noise and control parameter using a fractional factorial two level design, thus receiving the mean effect of this parameter on the target.

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Table 1. Noise parameter and their nominal values with corresponding variation. Noise Parameter Nominal Value Variation Comment Time series with hourly Heat load ±20% resolution Electricity refund 2.3 EUR ct/kWh Mainly influenced by politics ±30% Gas price 2.55 EUR ct/kWh Traded on stock market ±25% CO2-emission factor for Equal for electricity demand and 550 g/kWh ±20% electricity feed COP of GSHP 3.5 ±29% Nom. th. efficiency CHP 𝑓(𝑛𝑜𝑚. 𝑝𝑜𝑤𝑒𝑟) ±10% Nom. El. Efficiency CHP 𝑓(𝑛𝑜𝑚. 𝑝𝑜𝑤𝑒𝑟) ±5% Nom. Th. Efficiency NGB 98% ±5% Thermal Loss SES 0.65 W/(m²K) ±10% The ANOM of each noise parameter on the heat price as well as the CO2-emissions is shown in Fig. 4:

Fig. 4. Effect of different noise parameters on heat price (left) and CO2-emission (right). A high absolute effect of a noise parameter represents a high change of mean target value when changing the noise parameter from its lower to its upper value. 8

The results of the ANOM show that the heat load has the highest (negative) influence on the heat price. This can be explained by the performance of the CHP which feeds all the generated electricity into the grid, thus the more heat is generated by the CHP the lower the heat price. Continuatively, increasing CO2-emissions can then be linked to the higher heat load, leading to a higher demand for natural gas. Because of the electricity that the CHP can generate, leading to higher amounts of substituted CO2-emissions, this effect in turn is reduced by the CO2-emission factor, which will in fact reduce the CO2-emissions when at a high level. Also, the gas price has a high influence on the heat price and no influence on the CO2-emissions. The COP of the GSHP as well as the electrical efficiency of the CHP also affect (decrease) the CO2-emissions because of less electrical power and more electrical feed to the grid, respectively. In summary, the noise parameters included in the robust optimization are the heat load, the gas prices (because of the high influence on the heat price) as well as the CO2-emission factor and both, the COP and the electrical efficiency of the CHP. The control parameters are the dimensions of the potential supplier. To ensure structural optimization, those parameters have a lower bound of 0, which is equal to not being included in the system. The control parameters with their corresponding lower and upper bounds are listed in Table 2. Table 2. Control parameter and their corresponding lower and upper bounds. Control Parameter Lower Bound Upper Bound Area of STC 0 3000 𝑚² SES Volume 0 600 𝑚³ Nom. th. Power CHP 0 1000 𝑘𝑊 Nom. th. Power GSHP 0 1000 𝑘𝑊

4. Optimization Results The methodology described in Chapter 2 is applied to the district heating simulation model outlined in Chapter 3, where, in addition, the control parameters as well as the noise parameters that are included in the process of robust optimization are summarized. Step I: Approximation of Heat Price and CO2-Emissions Combining Control and Noise Parameters The initial design for sampling both, control and noise parameters, is a fractional two level design of resolution four with one central point, thus yielding 33 factor level combinations to sample the 9dimensional design space. As for the random ANNs topology, a maximum of one layer (for low complexity) with a maximum of 30 neurons are used for training. After simulating all 33 initial factor level combinations, 400 different ANNs are trained to approximate the heat price and the CO2emissions. The convergence criterion is the coefficient of determination R² on validation data (randomly chosen from 30% of the data) with a threshold value of 0.995. If the best ANNs out of the 400 trained networks fulfill the convergence criterion for both targets, the procedure stops, otherwise 10 new factor level combinations are added utilizing a MiniMax Latin Hypercube Sampling. For the whole process a total of 153 simulations needed to be carried out. Both approximations are optimized as stated in (1), the results of the deterministic optimization are shown in Table 3: Table 3. Results of the deterministic Optimization (Step I). Area of SES Nom. th. Power Target STC Volume CHP Heat Price 0 m² 0 m³ 0 kW CO2-Emissions 723 m² 159 m³ 0 kW 9

Nom. th. Power GSHP 0 kW 1000 kW

Target Value 7.98 ct/kWh 38.1 t/a

Step II: Approximation of Mean and Standard Deviation for Robust Optimization The initial inner Array (IA) is again chosen to be a fractional factorial two level design with one central point, thus a total of 33 simulations are initially carried out. Since the outer Array (OA) varies the noise parameters and is thus used to approximate the mean and standard deviation and, at the same time, has the highest influence on the total number of evaluations, it is required to have a resolution high enough for a good approximation of the robustness targets and low enough to reduce computational efforts. For this setup of noise parameters, a factorial design with two levels is considered to be sufficient. If the number of noise parameters increases, fractional factorial designs need to be considered due to the exponential growth of total numbers of evaluations. After approximating the mean and standard deviation of each setting of control parameters, four ANNs are trained with the same conditions as listed in Step I but with the convergence criterion being set to R²=0.999 due to the faster evaluations. This is also the reason why 20 (instead of 10) new evaluations are performed utilizing a maximin Latin Hypercube Sampling. The process comes to a halt after a total of 333 evaluations of different factor level combinations. Utilizing (4), a Pareto optimal robust optimum can be found (see green rectangle in Fig. 5) for both targets which then can be compared to the deterministic optimum (see red dot in Fig. 5) using (1) by placing the results into the Pareto front using (2) and (3). The latter was created only for visual comparison, but could also be used to assign meaningful weights to (4). The Pareto front in Figs. 5 and 6 are calculated applying the well-known NSGA-II algorithm [16]. Fig. 5 shows the results of this comparison for both targets. It its notable that the Pareto fronts are generated using the actual simulation with the results obtained by the optimization procedures.

Fig. 5. Comparison of the deterministic optimum with the robust optimum for the heat price (left) and the CO2-emissions (right). It is immediately apparent that the deterministic optimum for the heat price is part of the Pareto optimal solutions by finding the factor setting that evokes the best expected value. Nevertheless, the robust optimum is also part of the Pareto optimal solutions but compared to the deterministic optimum, it enforces a setting that reduces the standard deviation. In fact, the robust optimum reduces the normalized standard deviation by about 53% while only reducing the normalized mean by about 11%. In contrast, however, the deterministic and robust optimum for the CO2-emissions differ from each other. In fact, the deterministic optimum is, in terms of expected value and standard deviation, far away from the border to the non-realizable solutions (i.e. the Pareto front).

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The results indicate that the outcome of the robust optimization leads to useful results that, when combining both targets using (4), can be used to reduce the four-dimensional problem shown in Fig. 6 and to evaluate robust Pareto optimal solutions:

Fig. 6. Robust Pareto front of CO2-emissions and heat price combing (4) for the Heat Price and CO2-emissions.

6. Conclusion and Outlook In summary, it can be seen that the presented method has a high intuitive character and allows for a very intuitive robust optimization. At the same time the method allows for a deterministic (multiobjective) optimization as well as further analysis, where parameters and boundary conditions (of the analysis) can be varied easily. This is accomplished through the use of ANNs with a high generalization ability and the iterative modeling to ensure an optimal number of simulations. If the main aim is only to find the robust optimum for one target, this methodology can be adapted so that not the whole design space is sampled, consequently decreasing the number of expensive simulations necessary. Since the number of simulations represents the highest computational effort, reducing the convergence criterion of step 1 mainly affects the computational efforts. Alternatively, by using time series compression as shown in [17] and [18] the computational effort can be reduced and, moreover, the robust optimization might be accessible to a non-scientific audience as well. Future work will have to deal with a detailed comparison with other methods.

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