Analysis part one: Robustness evaluation

The first part analysis of the methodology is focused on the total energy usage of different refurbishment solutions. The final goal is to assess the refurbishment in terms of robustness to climate change.

The problems that we want to address and solve are: • How buildings behave in future climate

• How to assess the robustness of a particular refurbishment solution

• How to identify which solution is better in terms of robustness to climate change The first problem is the easiest to solve, at least theoretically. It is possible to calculate the energy usage of a particular building in any climate condition. However, these results are not completely reliable due to the intrinsic uncertainties of the weather files used as inputs as discussed in paragraph 4.4. We will address the others two problems by using these uncertainties related to future climate scenarios. A robust

solution should, in fact, be insensitive to climate change uncertainties. In practical

terms, we want to consider all the eighteen weather files that we have as the projection of future weather climates that could potentially happen in a “general future”. This fact means that, in this part of the analysis, it is not important if a weather file refers to a particular year or a particular scenario. The climate changes that the weather files predict are considered possible to the same degree. For this reason, the weather files referring to the present are considered in the robustness evaluation as well because they describe a possible stable climate. By considering all the weather files we have, a refurbishment solution is robust if the range of variation of energy usage is small. In other words, thanks to particular thermal properties of the envelope, the energy usage will be the same or will have little variations in many possible future climates.

BOX UPPER WHISKER LOWER WHISKER Q1 (25th PERCENTILE) Q3 (75th PERCENTILE) Q2 (50th PERCENTILE) MAXIMUM MINIMUM IQR 1.5 IQR 1.5 IQR Q1 (25th PERCENTILE) Q3 (75th PERCENTILE) Q2 (50th PERCENTILE) Q3 + 1.5 IQR IQR 1.5 IQR 1.5 IQR Q1 - 1.5 IQR OUTLIER OUTLIER

Figure 4.14: The box whisker plot in two different configuration of the whiskers.

The box whisker plots provide a useful way to compare distributions between several groups or sets of data without making any assumptions of the underlying statistical distribution. They use the quartiles of a group of data in order to analyse the distribution of the response to particular variations. The body of the box whisker plot consists of a “box” (hence, the name), which goes from the first quartile (Q₁), or 25th percentile, to the third quartile (Q3), or 75th percentile. Their difference is called interquartile

(IQR) and is represented by the equation Q₃ - Q₁. The line inside the box represents the second quartile (Q2), or 50th percentile, which is the median of the distribution.

The vertical lines (called whiskers) represent the variability outside the first and third quartiles and have two meanings. In general, they are the lowest datum still within 1.5 IQR of the lower quartile (Q1), and the highest datum still within 1.5 IQR of the upper

quartile (Q₃). This means that the whiskers represent the minimum and maximum of the data set if they are included in an arbitrary range based on the quartiles. If some values of the data set are higher or lower than the interquartile, the whiskers represent the boundary of the interquartile multiplied by 1.5 and subtracted from and added to the first and third quartile respectively. In this case, the values higher or lower than the whiskers are considered as outliers and are plotted separately. Figure 4.14 shows the two meanings of the whiskers, on the left when there are outliers and on the right when the whiskers are the maximum and the minimum of the distribution.

In our analysis, box whisker plots represent the base case and the twenty two refurbishment solutions. The boxes and whiskers for each thermal model, include all the energy usage from the eighteen weather files. The robustness of a particular refurbishment can be assessed with the dimension of the boxes. If the box is wide, the response of the building to climate changes varies substantially and, therefore, the particular refurbishment solution is not insensitive nor robust to climate variations. On the other hand, if the box is short, the thermal properties of the envelope make the building insensitive and robust to climate changes, no matter which climate and scenario there will be. In statistical terms, if the dispersion of the data around a certain value is small, the given strategy has a great robustness to climate variability because its thermal performance is stable.

The final results of this first part of the analysis are three box whiskers plots for each latitude, one for the heating, one for the cooling, and one for the total energy usage.

The box whiskers plots are useful to determine if there are big differences between the cases in term of robustness, but they are not easily readable if the variations are small. For this reason we introduce the Refurbishment Index (RI). The RI permits the evaluation of each refurbishment option by taking into account the width of the box (IQR) and the standard deviation of the refurbishment strategy in comparison with the same parameters of the base case. Before being able to use the RI, it is necessary to test if the set of data is normally distributed around the media. Only if the population is normally distributed, in fact, we can use the mean and the standard deviation to assess the dispersion of the values and calculate the RI.

Creating the Robustness Index (RI)

The process begins with the evaluation of individual values for each case, with respect to the standard deviation and the interquartile range. Each parameter is given a weight and then they are summed. The goal of the weighted sum approach is to give more importance to the dispersion around the mean, which is the standard deviation and determines the actual robustness of the refurbishment solution, by taking into account all the energy usage from all the weather files. The interquartile range, in fact, excludes the future climates that are out of the box. It is in any case useful to consider the interquartile range in the calculation for two reasons. The first one is that it indicates the width of the boxes, hence the dimension of the graphic robustness evaluation. The second reason it that it gives less importance to the outliers of the data set, that in statistical terms, should be less probable.

The individual value for each case is called comparison number and it is calculated with the following equation:

ωi= 0.3 · IQRi+ 0.7 · v u u u t n P j=1 (xj− µi)2 n (4.6) where:

- ω_i is the comparison number for the i-case (kW h

m2 )

- µ_i is the average energy usage for the i-case (kW h

m2 )

- IQRi is the interquartile range of the box whisker plot for the i-case (

kW h m2 )

- xj is a value of the energy usage for the j-weather file (

kW h m2 )

In our analysis, n, which is the number of data for each refurbishment, is eighteen. The computation of the interquartile excludes the outliers and the use of the standard deviation represents the mean difference of all the values from the mean, i.e., the dispersion of data.

The comparison number is calculated for both base case (BC) and refurbished cases (RC). In order to make the results between the seven latitudes comparable, the comparison number of each refurbishment must be normalized with that of the base case. This is the equation used for the calculation of the RI:

RIRCk = 1 − _ω RCk ωBC  (4.7) where:

- RIRCk is the Refurbishment Index for the k-refurbishment (unit-less)

- ω_RCk is the comparison number for the k-refurbishment (kW h

m2 )

- ωBC is the comparison number for the base case (

kW h m2 )

The higher RI a refurbishment case has, the better the refurbishment is in terms of robustness to climate change. This is because the comparison number measures two parameters that must be small to describe a robust refurbishment solution, the interquartile and the standard deviation. A small interquartile, hence a small box, indicates that the majority of the energy usage is close to the median, and excludes outliers. A small standard deviation affirms that the majority or all the values have a small distance from the average energy usage, i.e., the dispersion of data is small. The comparison number, therefore, must be a small value if the refurbishment is a good solution in terms of robustness to climate change. If this is the case, the comparison number for the refurbishment k is lower compared to the one of the base case. As a consequence the normalized comparison number (ωRCk

ωBC

) is a small number and the RI is a high value.

The RI is useful to calculate a value which indicates the robustness of a partic- ular refurbishment solution but it is also valuable in the comparison of the thermal

performances between the base case and the refurbishments. These are the general rules:

• If ωRC > ωBC then RI < 0, which means that the refurbishment solution is less

robust than the base case

• if ωRC < ωBC then RI > 0, which means that the refurbishment case is more

robust than the base case.

The RI can range from −∞ (if the ωBC is quite low and the ωRC is really high) to 1

(best score achievable if the ω_BC is quite high and the ω_RC is really low). In general, with the energy usage values typical of Europe, the RI ranges from -1 to +1. A negative value indicates that the refurbished building is characterized by more dispersed values for cooling and heating energy usage than those of the base case.

The RI can be calculated for the heating, the cooling and the total energy usage. This latter is useful in the comparison of the same solution at different latitudes to see what global warming means in different climatic conditions and for several refurbishment interventions.

The RI does not take into account the energy usage. A refurbishment could have a high RI value, and so be robust to climate change, but at the same time could have very high energy usage. This could happen if all the values are concentrated around a very high energy usage. For this reason it is necessary to introduce another index which is able to classify the refurbishments in terms of energy usage. The index is described in the following subsection.