Application of sensitivity analysis in building energy simulations: combining first and second order elementary effects Methods

David Garcia Sanchez; Bruno Lacarrière; Marjorie Musy; Bernard Bourges

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Application of sensitivity analysis in building energy simulations: combining first and second order elementary effects Methods

David Garcia Sanchez, Bruno Lacarrière, Marjorie Musy, Bernard Bourges

arXiv:1203.3055v2Mar 14, 2012cs.CEstat.AP

TL;DR

Building energy models use many diverse input variables, making their influence on outputs difficult to assess. This study combines first- and second-order Morris elementary-effects analyses in ESP-r and finds that higher-order analysis identifies nonlinearities and interactions that support first-order results.

Problem
Building energy models involve many diverse input variables, complicating assessment of their influence on outputs.
Method
The study combines first- and second-order Morris elementary-effects sensitivity analyses with the ESP-r building energy model.
Results
Higher-order analysis identifies possible nonlinearities and interactions, helping support interpretation of first-order sensitivity results.
Takeaways & Limitations
Combining sensitivity-analysis orders provides a broader assessment of complex building-model behavior than first-order analysis alone.
Takeaways & Limitations
Sensitivity results are not general across modeling tools and depend on the model and the chosen input-variation range, so each case requires separate analysis.

Abstract

from arXiv · show

Sensitivity analysis plays an important role in the understanding of complex models. It helps to identify influence of input parameters in relation to the outputs. It can be also a tool to understand the behavior of the model and then can help in its development stage. This study aims to analyze and illustrate the potential usefulness of combining first and second-order sensitivity analysis, applied to a building energy model (ESP-r). Through the example of a collective building, a sensitivity analysis is performed using the method of elementary effects (also known as Morris method), including an analysis of interactions between the input parameters (second order analysis). Importance of higher-order analysis to better support the results of first order analysis, highlighted especially in such complex model. Several aspects are tackled to implement efficiently the multi-order sensitivity analysis: interval size of the variables, management of non-linearity, usefulness of various outputs.

1. Introduction

Building energy models couple multiple phenomena through many diverse inputs, making sensitivity analysis useful for ranking parameters, identifying interactions, and guiding model simplification. The study combines ESP-r with the Morris method and its second-order extension to analyze parameter influence and interactions.

Motivation: Building energy models couple phenomena such as occupancy, micro-climate, envelope, and HVAC through many diverse input variables.This complexity motivates methods that can assess relative parameter influence and interactions.
Motivation: Sensitivity analysis ranks input parameters or parameter families according to their influence on model outputs and modeling objectives.It can support building design, archetype definition, model development, simplification, and assumption validation.
Model development: A detailed upstream model combined with sensitivity analysis can identify important variables and parameter couplings for defining a simplified model.Selecting the most important variables helps determine the simplified model’s structure.
Study contribution: The study combines ESP-r with the Morris method and a second-order extension for analyzing interactions between parameters.The paper reviews sensitivity-analysis methods, describes the elementary-effects variants, tests them on an apartment building, and discusses their results.

2. Background

Sensitivity analysis methods have been studied for decades and applied across many sectors. During this period, researchers developed new methods and improvements offering different solutions depending on the objective.

2. Background: Sensitivity analysis methods have demonstrated their strength across many sectors over decades.
2. Background: New methods and improvements have been developed to offer different solutions depending on the objective.
2. Background: Hamby proposed an inventory dividing parameter sensitivity techniques into three categories, including methods assessing individual-parameter influence.
2. Background: Differential Sensitivity Analysis and one-at-a-time sensitivity measurements are examples of methods assessing individual parameters.

sures, Factorial Design, Sensitivity Index, Importance Factors, and Subjective

Sensitivity analysis methods range from local derivative-based approaches to global methods that vary inputs across wider domains. The Morris method offers an efficient intermediate screening approach, but its higher-order extension had not yet been applied to building energy simulations.

Sensitivity-analysis methods: Local methods estimate partial derivatives for limited input values, whereas global methods vary parameters across a wider domain.Global methods may be quantitative or qualitative.
Morris method: The Morris method screens large parameter sets using elementary effects, whose means and standard deviations distinguish influential, negligible, linear, and non-linear influences.It is derived from one-factor-at-a-time screening and uses sampled points on a regular grid.
Morris method: The Morris method is model-independent and provides a compromise between accuracy and efficiency, although its applications remain limited.Variance-based methods better distinguish non-linearities and interactions but require substantially more model evaluations.
Computational cost: 14,000 runs are required for variance-based analysis of a 12-input model, about one hundred times the cost of first-order Morris and ten times second-order Morris.The comparison highlights the lower computational cost of Morris screening and its higher-order extension.
Building-energy applications: Building-energy applications used Morris screening to identify influential parameters, including 12 of 81 factors explaining 94% of overheating-hour variability.Other studies found five of 129 factors responsible for most energy-rating uncertainties and identified key thermal-comfort parameters.
Research gap: Second- and upper-order Morris analysis had not yet been applied to building energy simulations despite its stated advantages and low computational cost.The gap motivates extending Morris analysis beyond first-order screening in this domain.

3. Methodology

The methodology applies Morris elementary-effects sensitivity analysis to a normalized, discretized building-energy model, extending it to second-order effects to classify parameter importance and interactions. Experiments use ESP-r simulations of a complex multi-zone residential building with specified parameter grids and analysis sets.

Morris elementary-effects method: Input variables are transformed to dimensionless values in (0, 1), forming a unit-length hypercube H_k with discretized grid values.The model is represented as y(x), where x contains k real input variables defined over continuous intervals.
Morris elementary-effects method: Each Morris trajectory contains k + 1 points, changes one parameter once by a predefined step, and evaluates the model at every point.Multiple random trajectories provide repeated elementary-effect estimates for each input at a cost of r × (k + 1) simulations.
Morris elementary-effects method: The average absolute elementary effect ranks input importance, while its standard deviation indicates non-linearity or interactions with other parameters.Using absolute effects avoids cancellation in non-monotonic models but loses effect-sign information.
Second-order extension: Second-order elementary effects characterize the influence of changing two factors together, with an optimized computational requirement of about k^2r model evaluations.Statistics of second-order effects can be interpreted analogously to first-order effects, while first-order variability also informs higher-order interactions.
Second-order extension: Combining first- and second-order analyses at the same intervals classifies parameter importance across first, second, and higher orders and supports interpretation of complex models.The combined approach can also inform reduced models based on the most important parameters.

4. Results and discussion · 4.1. General remarks on the presentation of results

The results are presented with elementary-effects scatter plots that show each input factor’s absolute average and standard deviation. Ratios involving these measures help classify linearity, monotonicity, non-linearity, and possible interactions, while also providing checks on sensitivity-analysis results.

4.1. General remarks on the presentation of results: Scatter plots represent each input variable with absolute average µi∗ on the x-axis and elementary-effects standard deviation σi on the y-axis.These plots are used to present the elementary-effects analysis of building thermal simulations.
4.1. General remarks on the presentation of results: The ratio σi/µi∗ complements µi∗ by indicating whether an input factor’s effect is approximately linear, monotonic, or non-linear.The ratio is introduced as a complement to the absolute average, with its interpretive use justified through the elementary-effects distribution.
4.1. General remarks on the presentation of results: Under a normal assumption, 95% of elementary-effects estimates fall within µi ± 1.96 σi.This statistical property supports interpreting the dispersion of elementary effects around their mean.
4.1. General remarks on the presentation of results: When σi/µi is below 0.10, effects are nearly constant and the input has an almost linear model effect; below 0.5, the response is considered monotonic.The thresholds are based on the stated normal-distribution interpretation of elementary-effects estimates.
4. Results and discussion: Observed scatter shows monotonic behavior for σi/abs (µi) below one, whereas σi/abs (µi) above one indicates non-monotonic behavior.This empirical comparison broadens the monotonic range beyond the initially expected threshold of 0.5.
4.1. General remarks on the presentation of results: Plotting lines with slopes σ/µ∗=0.1, 0.5, and 1 divides factors into linear, monotonic, almost monotonic, and strongly non-linear or interacting zones.Factors above σ/µ∗> 1 are associated with marked non-monotonic non-linearities or interactions with other factors.
4.1. General remarks on the presentation of results: The four-zone classification provides a way to check sensitivity-analysis results against the expected physical behavior of the model.The text presents this comparison as a check when sensitivity results contradict physical understanding.
4.1. General remarks on the presentation of results: Alternative plots using horizontal lines to distinguish linear, monotonic, and highly non-linear domains convey the same information and can be easier to read.This presentation is used for some figures, including Figs. 5c, 5d, 6c, and 6d.

4.2. Computational experiment A: first-order analysis results

First-order Morris analysis identified building size as the dominant influence on yearly heating demand, followed by set point temperature, ventilation rate, and insulation thickness. Results were broadly consistent across transformed and alternative outputs, while comfort sensitivity differed and highlighted external temperature, geometry, and glazing orientation.

Yearly heating load: 470 000 kWh/year was the average elementary effect for changing building height from 9 m to 27 m.This illustrates the influence of a full-scale change in a dimensional input.
Heating load per cubic meter: Annual heating load per cubic meter reduced the size effect and made set point temperature, ventilation rate, and insulation thickness predominant.Set point temperature and ventilation rate showed behavior closer to linear, while dimensional parameters remained non-negligible and contributed to variability.
Logarithmic transformation: The logarithmic transformation preserved the important-parameter ranking while reducing σ/µ∗ ratios for temperature-related parameters to close to 0.1.Glazing ratio, rotation, and insulation thickness remained non-linear after transformation.
Alternative outputs: Comfort sensitivity differed from heating-demand sensitivity: external temperature was most important, while environmental and geometric parameters followed, including side-B glazing ratio.The Morris method could therefore help designers assess building impacts on comfort and examine sensitivity to solar gains.

4.3. Computational experiment B: first- and second-order results

Experiment B combined first- and second-order Morris analyses across 12 selected parameters, two interval sizes, and two heating-related outputs. Second-order interactions were often substantial, especially among building-size parameters, while reducing intervals shifted most effects toward linear behavior and changed the dominant influence to set-point temperature.

Experimental design: Second-order analysis required 5760 runs with r = 10, so 12 of the 24 initial parameters were retained after omitting negligible ones.The study used large and small parameter intervals and analyzed annual heating needs plus their natural logarithm per cubic meter.
Second-order interactions: Almost 0.2 was the average second-order elementary effect for interaction (3;8) with the logarithmic response, exceeding several important first-order effects.For annual heating needs, interaction (3,2) exceeded first-order averages for every parameter except 2, 3, and 4.
Small-interval analysis: Reducing parameter intervals by a factor 10 placed almost all first- and second-order effects in the linear zone, with set-point temperature becoming most influential and insulation thickness nearly irrelevant.The reduced-interval analysis again found set-point temperature with the highest first-order average and standard deviation, while insulation thickness had almost no output influence.

5. Conclusions

The study combines first- and second-order Morris sensitivity analyses for an ESP-r building simulation. It shows how output transformations and interaction analysis clarify parameter importance, while emphasizing that conclusions depend on the modeled situation and variation intervals.

Conclusions: The combined Morris approach identifies parameter families, non-linearity, and higher-order interactions even without a precise parameter ranking.First-order analysis groups inputs by importance and flags cases requiring higher-order investigation.
Conclusions: Output forms including kWh/year, kWh/year.m3, and ln(kWh/year.m3) help reduce or explain non-linearity and interactions.Specific values per m3 or m2 reduce correlation with size-related parameters, while logarithmic outputs help identify non-linearity origins.
Conclusions: Second-order analysis sorts high-standard-deviation variables and specifies their pairwise interactions, with its usefulness amplified across output forms.The same simulation trajectories support multiple outputs; complementary post-processing is sufficient for transformed outputs.
Conclusions: The proposed Morris presentation classifies parameters or parameter pairs as linear, monotonic, almost monotonic, or highly non-linear/interaction-of-higher-order.This classification provides a new way to present elementary-effects sensitivity results.
Conclusions: Sensitivity results are situation-specific: they depend on the input parameters, their variation ranges, modeling goals, and carefully chosen intervals.The conclusions are valid only for the particular building, model, and analysis situation studied.

Tables

The tables describe the apartment building test and list the parameters and intervals used in the first- and second-order Morris-method experiments.

Table 1 presents the main characteristics of the apartment building test.
Table 2 lists the parameters and different intervals used in the first- and second-order Morris-method experiments.

Figures

The figures document the apartment-building case study and visualize first- and second-order elementary effects across comfort, power, and annual-heating outputs. They include scatter plots and estimated means and standard deviations under large parameter intervals with r = 10.

Case study: Fig. 1 presents the apartment building case study.
First-order analysis: Fig. 2 plots σ_i/µ∗_i against σ_i/abs(µ_i) for all first-order analyses.
First-order analysis: Fig. 3 estimates the absolute average µ∗ and standard deviation σ of first-order effects.
Outputs: The figures analyze elementary effects for summer mean temperature, power exceeded during 1000 hours/year, and annual heating needs.The temperature output is a comfort factor, while the heating analyses distinguish first- and second-order effects.
Annual heating needs: Annual-heating figures compare first- and second-order effects using a large interval for each parameter and r = 10 elementary effects per parameter.
Multi-order effects: Figs. 5 and 6 estimate first- and second-order elementary effects for two outputs.