Experimental design and response surface methodology in energy applications: A tutorial review

Energy Conversion and Management 151 (2017) 630–640

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Review

Experimental design and response surface methodology in energy applications: A tutorial review

MARK

Mikko Mäkelä Swedish University of Agricultural Sciences, Department of Forest Biomaterials and Technology, Division of Biomass Technology and Chemistry, Skogsmarksgränd, 90183 Umeå, Sweden Aalto University, School of Chemical Engineering, Department of Bioproducts and Biosystems, PO Box 11000, 00076 Aalto, Finland

A R T I C L E I N F O

A B S T R A C T

Keywords: Design of experiments Multiple linear regression Process modeling Process optimization Variance

Experimental design and response surface methodology are useful tools for studying, developing and optimizing a wide range of engineering systems. This tutorial provides a summary and discussion on their use in energy applications. The theory and relevant calculations are clearly presented and discussed along with model diagnostics and interpretation. This is followed by a review of recent reports within the energy field. Overall, this contribution will clarify many aspects of experimental design and response surface methodology that are often confusingly discussed in the academic literature and summarizes relevant applications where they have been found useful.

1. Introduction Experimental design is a collection of tools used for studying the behavior of a system. Experimental design, or design of experiments, involves planning and performing a set of experiments to determine the effects of experimental variables on that system. The acquired data is separated into variation generated by the system itself and respective uncertainties or errors always present in empirical data. A statistically valid model is obtained, which by definition contains information on the effects of experimental conditions on the direction and magnitude of the measured response. The required experiments are also performed in a way that maximizes the information that can be extracted from a limited number of experiments. Once a satisfactory model has been determined, it can be used for predicting future observations within the original design range. Experimental design is thus useful for, not only studying, but also developing and optimizing a wide range of engineering systems. The method was originally developed by Fisher in the 1930s through factorial designs and analysis of variance for agricultural and biological research [1,2]. Response surface methodology was first discussed in the 1950s by Box and Wilson within chemical experimentation, and generally includes mathematical and statistical tools for both the design and analysis of response surfaces [3–5]. In practice, the methods are today closely related and the use of response surface methodology is without exception based on experimental designs. In this work, experimental design is used to refer to practices included in both topics. Experimental design is closely related to the mentality of learning

by experience and sequential experimentation. The effects and statistical significance of a larger group of experimental variables can be determined through factorial or screening designs, which enable choosing the relevant variables or conditions for the next set of experiments. Experimental designs are constructed in a way that eliminates or minimizes correlations between the chosen variables. This allows independent estimation of variable effects and their potential interactions. Here lies an important advantage of experimental design, as the variables are not varied one at a time while the others are being held constant. This approach assumes that the variables do not interact, i.e., the effect of one variable stays the same even though the others change. In many situations, this assumption can be unjustified. As an example, increasing the concentration of a catalyst might lower the temperature required for producing bio-oil of a specific quality. The effect of temperature thus changes based on catalyst concentration, indicating that the two variables interact. Although experimental design is useful in many areas of energy research, it has no natural connection to the studied system. What is obtained is a simple mathematical approximation of the response based on empirical data. More simple designs and models are often easier to interpret, which increases their value in practical situations. The chosen design also determines the level of detail and complexity that can be described with the subsequent model. Factorial designs can be used for quantifying linear and interaction effects, whereas optimization designs allow describing more complex behavior by including higher order model components. The mathematical and statistical procedures of experimental design

E-mail addresses: [email protected], mikko.makela@aalto.fi. http://dx.doi.org/10.1016/j.enconman.2017.09.021 Received 25 May 2017; Received in revised form 18 August 2017; Accepted 8 September 2017 0196-8904/ © 2017 Elsevier Ltd. All rights reserved.

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have been well documented (e.g. [6–9]). In addition, several reviews have been published especially within analytical chemistry and food engineering [10–20]. As an example, Bezerra et al. [14] provided a good overview for the optimization of analytical methods and Leardi [15] included a discussion of mixture design for more specialized applications. Experimental design and response surface methodology are also frequently discussed in journals such as Technometrics published by the American Statistical Association [21]. What is missing is a practical tutorial that summarizes relevant reports on the use of experimental design within energy applications. This contribution aims to fulfill that knowledge gap. The theory and relevant equations are clearly presented and discussed along with model diagnostics and interpretation. The calculations are also illustrated based on an imaginary data set, which allows the reader to follow through the thought process. This is followed by a review of recent reports within the energy field. Overall, this contribution will clarify many aspects of experimental design that are often confusingly discussed in the academic literature and summarizes relevant applications where it has been found useful. 2. Materials and methods This section describes the data set, relevant model calculations and diagnostics along with details on data compilation and review. Modeling results are then presented and discussed separately in Sections 3 and 4. The discussed calculations are also illustrated based on the data set. The calculations were performed using the Matlab R2016a (The Mathworks, Inc.) software package, but can be performed with any software capable of linear algebra. Open source alternatives are also available. Data plotting was performed with the OriginPro 2015 (OriginLab Corp.) software package. Once the modeling results have been presented and discussed, recently published work is reviewed to illustrate and discuss practical examples from the energy field.

Fig. 1. Some common designs in two dimensions; (a) a factorial design, (b) a central composite design with a coded star-point distance α = 1, (c) a central composite design with α > 1 and (d) a Box Behnken design.

full factorial design on two variables. Factorial designs generally contain only two variable levels, a minimum and a maximum, and are described with the abbreviation 2k, where k denotes the number of variables in the design. Experiments 5–8 are called star-points and are added to central composite designs for calculating higher order model components. Their distance α is expressed in coded units from the design center and is generally dictated by the problem at hand. Setting α as 1 or 4 2k are secure choices. The value 4 2k guarantees rotatability, i.e., spherical prediction variance around the design center. Experiments 9–11 are replicated center-points and enable estimating true replicate error. Central composite designs were first introduced by Box and Wilson in 1951 [3] and together with Box Behnken designs [22] have become one of the most common designs used for quadratic models. In general, many different designs are available in the literature, both for screening and optimization purposes, and will not be discussed here. The interested reader is recommended to turn to the many books or commercial software available in the field. Some common designs in two dimensions are illustrated in Fig. 1. The same logic applies in three or more dimensions.

2.1. The data set The data set describes experiments that were performed to determine the effect of temperature and catalyst concentration on the molecular weight of bio-oil (Table 1). The data set was kept small to maintain simplicity. The temperature was varied within 160–320 °C and the catalyst concentration within 0.2–0.8%. The molecular weights of the attained oils were chromatographically determined and were in the range 0.59–2.0 kg mol−1. A lower molecular weight was favorable to increase the performance of the oil in subsequent applications. The experimental order was randomized to minimize systematic errors. The experiments were organized according to a face-centered central composite design with two variables or design factors and three replicated center-point experiments. A total of 11 experiments were performed. As illustrated in Table 1, the design included three levels for each variable and can be used for quantifying linear, interaction and higher-order model terms. The first four experiments in Table 1 equal a

2.2. Coding and model coefficients Modeling is based on approximating the true behavior of a response:

Table 1 The data set based on a central composite design with two variables. Experiment

Temperature (°C)

Catalyst (%)

Molecular weight (kg mol−1)

1 2 3 4 5 6 7 8 9 10 11

160 320 160 320 160 320 240 240 240 240 240

0.2 0.2 0.8 0.8 0.5 0.5 0.2 0.8 0.5 0.5 0.5

2.0 0.85 1.8 1.0 1.7 0.59 1.4 1.2 0.89 1.2 0.94

y = f (ϕ1,ϕ2,⋯,ϕk ) + ε

(1)

where y denotes the measured response as a function of (ϕ1,ϕ2,…,ϕk ) variables and other sources of variability ε . The variable values are coded to compare their effects within the design range:

xi =

(ϕi−ϕmin ) −1 Δϕ/2

(2)

where xi denotes a coded value and ϕi , ϕmin and Δϕ the respective variable value, minimum variable value and variable range, all in original units. In this way, the factorial design points in Table 1 range from −1 to 1 and the design center is situated at (0, 0). A quadratic regression equation is generally used to approximate y: 631

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k

y = β0 +

∑

βi x i +

i=1

k

k

∑ ∑ i=1

βij x i x j +

j=i+1

∑

βii x i2 + ε

i=1

Table 2 The general structure of an ANOVA table in experimental design.

(3)

where β0 describes the average value of y, βi , βij and βii the first order, interaction and quadratic coefficients, respectively, xi the coded factors and ε the model residual. In matrix notation Eq. (3) becomes:

y = Xb + e

e = y−y ̂

Sum of squares (SS)

Mean square (MS)

F-value

Total corrected Model

n−1 k

SStot SSmod

MSmod

MSmod/ MSres

Residual Lack of fit Pure error

n–p n–p–(nr–m) nr–m

SSres SSlof SSpe

MSres MSlof MSpe

calculated as SSmod = SStot − SSres. Respective mean squares (MSs) or variances are then determined by dividing the sum of squares with the respective degrees of freedom. The effects of individual model coefficients can also be determined through their respective sum of squares. The general structure of an ANOVA table is given in Table 2. The main part of the ANOVA table is used for F-testing the model against the residuals. The null hypothesis is that all coefficients equal zero, H0: β1 = ⋯ = βk = 0 and the model basically equals noise. The alternative hypothesis H1 is that at least one β ≠ 0. The null hypothesis MS is rejected if MSmod > F∝ ,k,n − p . res An additional F-test can be performed on so-called lack of fit, a property that tests whether the model adequately describes the data. An estimate of true replicate error through center-points or otherwise replicated experiments is required. True replicate, or pure error can be calculated as:

where y ̂ denotes a n × 1 vector of predicted values:

y ̂ = Xb

(6)

It is convenient to minimize the sum of squares of model residuals, which is a scalar:

∂e′e (y−Xb)′ (y−Xb) = …=−2X′y + 2X′Xb = 0 ∂b

(7)

→ b = (X′X)−1X′y

(8)

m

SSpe =

pe

R2 =

SSmod SS = 1− res SStot SStot

One weakness of the R value is that it will continue to increase as more terms are added to the model. This eventually leads to overfitting, i.e., explaining noise rather than meaningful information. An adjusted R2 can be calculated, which takes into account the number of terms in the model:

> t α2 ,n − p .

2 Radj = 1−

The statistical significance of a model can be tested by comparing the variation explained by the model and the variation of model residuals through the analysis of variance (ANOVA). The total sum of squares (SStot) of y is:

(yi −y )2

i=1

(10)

The sum of squares of model residuals e is calculated as:

n

n

SSres =

∑ i=1

SSres /(n−p) SStot /(n−1)

(yi −yi )̂ 2

(14)

If the model is used for prediction, it is good to estimate its predictive capability. One way to do this is to take out an observation and build a new model based on the remaining n − 1 observations. The new model can then be used for predicting the removed observation. This procedure is sometimes called the leave one out method and is repeated until all observations have been predicted once. The prediction error sum of squares can be determined as [9]:

n

∑

(13) 2

2.3. Model diagnostics

SStot =

(12)

It is useful to describe how much of the original data variation the model explains. This can be done through the coefficient of determination, the R2 value. The R2 is calculated as:

where σ 2̂ is an estimate of model variance and Dii is the ith diagonal element of (X′X)−1. The statistical significance of each βi can then be determined based on a t-test with the null hypothesis H0: βi = 0 , which βi

(yij −yi )2

j=1

where nr denotes the number of replicates at a design location and m the number of design locations where replicates were performed. The lack of fit sum squares is then determined as SSlof = SSres − SSpe. The null hypothesis H0 of the lack of fit test is that the model has no significant lack of fit. As opposed to the first F-test on model significance, it is favorable for the lack of fit test to confirm the null hypothesis. The null MSlof hypothesis is rejected if MS > F∝ ,n − p − c,c , where c = nr − m.

(9)

se (βi)

nr

∑∑ i=1

where Eq. (8) is the least squares estimate of b, in which b itself is linear. The vector b can only contain as many coefficients as there are experiments in the design, otherwise the coefficient estimates will be biased and their effects cannot be separated. Correlation between the different columns of X can also lead to unstable estimates of b or a situation where X′X cannot be inverted. Using appropriate designs and avoiding excessive variation in planned variable levels will help to avoid such situations. After determination of the model vector, the predicted values y ̂ and residuals e can be calculated through Eqs. (6) and (5), respectively. The standard error (se) of each coefficient in b can be estimated through the covariance matrix:

σ 2̂ Dii

MSlof/MSpe

Subscripts; tot = total, mod = model, res = residual, lof = lack of fit, pe=pure error. MS=SS/df. p = k + 1.

(5)

should be rejected if |t0 | =

Degrees of freedom (df)

(4)

where y is a n x 1 vector of response values, X a n x p matrix of coded values, b a p x 1 vector of model coefficients and e a n x 1 vector of model residuals. X is generally called the design matrix and includes columns for determining the interaction and quadratic coefficients included in Eq. (3). In addition, a column of ones needs to be included in X for the determining the average value of y in the design center. A central composite design matrix with three center-point experiments for any two design variables xi and xj is illustrated in Table A.1 (Appendix A). Once the design matrix has been built, the model vector can be determined. The aim is to minimize the difference between the observed and predicted values, i.e., the model residual:

se (βi ) =

Parameter

SSpre = (11)

∑ i=1

2

⎛ ei ⎞ ⎝ 1−hii ⎠ ⎜

⎟

(15)

where hii denotes the ith diagonal element of the hat matrix

As the sum of squares are additive, the model sum of squares can be 632

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H = X (X′X)−1X′. The predictive capability of the model can now be described as: 2 Rpre = 1−

Table 3 ANOVA of the final model. The tabulated values have been rounded.

SSpre

An important part of model diagnostics is to look at the residuals. A common method is to normalize the residuals with an estimate of model error, σ ̂ (Table 2):

ei,N =

ei σ 2̂

=

ei MSres

(17)

Internally (ei,IS ) or externally (ei,ES ) studentized residuals can also be used:

ei,IS =

ei σ ̂ 1−hii

ei,ES = ei,IS

Parameter

Degrees of freedom (df)

Sum of squares (SS)

Total corrected Model Residual Lack of fit Pure error

10 3 7 5 2

2.0 1.8 0.14 0.09 0.06

(16)

SStot

n−p−1 n−p−ei2,IS

Mean square (MS)

F-value

p-value

0.61 0.02 0.02 0.03

30.4

< 0.01

0.62

0.71

design. The ANOVA table is illustrated in Table 3. 4. Discussion

(18)

4.1. The data set (19)

The first step after determining the model vector is to estimate the statistical significance of model coefficients. Coefficients which are found statistically insignificant should be removed from the model unless they are included in significant interactions or higher order coefficients. As illustrated in Fig. 3a, the initial model included coefficients for the full quadratic model presented in Eq. (3). Coefficients for the interaction effect of temperature and catalyst concentration and the quadratic effect of temperature (b12 and b11, Fig. 3a) were found statistically insignificant (p > 0.10) and were removed from the final model (Fig. 3b). If the model only contains linear components the coefficient values can be compared, otherwise it is better to look at the response surface for interpretation. Once the final model has been determined, it is useful to look at the residuals. If the residuals represent pure random error, they should lack clear structure and have a normal distribution with zero mean and variance σ 2̂ . The raw residuals however are not very useful and thus different normalization or standardization norms are often used. Some examples are given in Fig. 4. Residuals can be plotted based on run order (Fig. 4a). If the residuals are normally distributed, they should fall on an approximately straight line in the normal probability plot (Fig. 4b). Residuals normalized with an estimate of model error σ ̂ (Fig. 4c) should follow a normal distribution with zero mean and approximately unit variance. However, raw and normalized residuals are more likely correlated and have non-constant variance [23]. Internally studentized residuals have constant unit variance with similar statistical properties as the raw residuals, which depends on the random error distribution, the design and the number of performed experiments [23]. Taking into consideration the relatively low number of experiments often included in experimental designs, externally studentized residuals are appropriate as they have a t-distribution with n–p–1 degrees of freedom and take into consideration the effect of individual experiments on the predictive capability of a model. A good alternative is the Williams graph (Fig. 4d), where externally studentized residuals are plotted as a function of leverage within the design [23]. Although leverage is used for standardizing the residual, including it in the graph makes visual interpretation easier. Note that the horizontal lines in Fig. 4d are based on the Student’s t distribution with α = 0.10 and n–p–1 degrees of freedom. As illustrated in Fig. 4, experiment 4 was most susceptible to be an outlier but in this example was not removed from the final model. The obtained model can be used for predicting novel observations within the original design range. Extrapolation should be avoided, as the prediction variance increases dramatically. Predictions are best illustrated through a response surface, where the predicted response is plotted as a function of design variables in two or three dimensions. As illustrated in Fig. 5a, the response surface followed the shape of an ellipse due to the quadratic effect of catalyst concentration. The molecular weight of bio-oil was minimized at higher temperatures, as an

A good overview of residual diagnostics in regression modeling has been given by Meloun et al. [23] and will not be repeated here. Model residuals are further discussed in Section 4. 2.4. Data compilation and review Literature on the use of experimental design and response surface methodology was compiled from ScienceDirect [24] by searching for ‘energy’ and ‘experimental design’, ‘design of experiments’ or ‘response surface methodology’. Only recent papers were reviewed to provide an updated view. In addition, only papers relevant to the energy field are discussed. The results were not exhaustive, but provided a representative overview on the use of experimental design within the field. 3. Results A regression model for the molecular weight of bio-oil was successfully determined as a function of temperature and catalyst concentration, Fig. 2. Insignificant coefficients were removed from the final model (p > 0.10). The model was found statistically significant (p < 0.01) with no significant lack of fit (p > 0.10). The determined R2 value was 0.93 indicating that the model explained 93% of variation in the original data. The respective adjusted R2 value was 0.90. The R 2pre value determined by cross-validation suggested that the model could explain 82% of variation in predicting novel observations. As illustrated in Fig. 2, the model described the data reasonably well and could correctly predict approximately 4 out of 5 observations in the original

Fig. 2. Predicted vs. observed values for the final regression model. The 45° straight line illustrates a perfect fit.

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Fig. 3. Coefficients and respective 95% confidence intervals for (a) the full quadratic model, and (b) the final model after removal of insignificant coefficients. In the plot ‘b1’ denotes the coefficient for temperature and ‘b2’ the coefficient for catalyst concentration.

behavior of multiple responses can be studied by overlaying them on top of each other and different optimization and desirability routines have also been developed. When interpreting response surfaces, it is good to keep in mind that the plots are based on predictions. Each predicted point thus has a confidence interval. If an estimate of model error is known, prediction uncertainty or prediction variance can be determined [25–27]. In general, this uncertainty depends on the design, the final model and the prediction location. A response surface based on the half-fraction of the 95% confidence interval is illustrated in Fig. 5b [28,29]. As an example, a temperature of 280 °C coupled with a catalyst concentration of 0.5%

optimum catalyst concentration existed somewhere in the middle of the catalyst range. In this case a local optimum for catalyst concentration could be determined. Derivation of the final model equation in respect to catalyst concentration gave an optimum of 0.52%. In general, the character of a stationary point can be canonically determined by rewriting the general regression equation as y= b0 + x′b + x′Bx , where x = (x1,x2,…,xk )′, b = (β1,β2,…,βk )′ and B is a symmetric matrix that contains the pure second-order coefficients as diagonals and βij/2 as the ij elements [5]. The eigenvalues of B then indicate whether the stationary point is a minimum, a maximum or a saddle point. The interpretation however usually becomes complicated after 2–3 components. The

Fig. 4. Residuals based on the final model; (a) residuals based on run order, (b) a normal probability plot of the raw residuals, (d) residuals normalized with their average standard deviation (σ )̂ and (f) externally studentized residuals as a function of leverage. The dashed horizontal lines were plotted based on (c) the normal distribution α = 0.05, and (d) the Student’s t distribution, 6 degrees of freedom, α = 0.10. The dashed vertical line in (d) indicates the limit for high leverage experiments.

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Fig. 5. Response surfaces of (a) the predicted molecular weight of bio-oil (kg mol−1) and (b) the half-fraction of the respective 95% confidence interval.

would lead to a molecular weight of 0.81 ± 0.37 kg mol−1 with 95% certainty (Fig. 5a and b). One way to test the performance of the obtained model is to perform additional verification experiments. With multiple verifications a root mean squared error (RMSE) can be determined [30,31]:

experimental conditions on methyl ester and diesel yields from the transesterification of scum oil [43], mixed vegetable oil [44], oleander oil [45], sun flower oil [46] and waste cooking oil [47] have been determined using central composite, fractional central composite, Box Behnken and 33 designs. The effect of nanocatalyst composition on respective basicity and methyl ester yield has also been reported based on a central composite design [48]. Central composite designs have been used to determine the effects of esterification and transesterification of shea butter [49] and neem oil [50] on respective free fatty acid and diesel yields. Betiku et al. [50,51] and Ighose et al. [45] compared the performance of response surface models with artificial neural networks (ANN) and adaptive neuro-fuzzy inference systems (ANFIS) within biodiesel production. Although ANN and ANFIS provided better predictive ability, experimental design enables the determination of variable effects on the measured response not provided by ANN or ANFIS. In general, a degree of nonlinearity in response data can be described using higher order model components or response transformations. The performance of 42 and central composite designs in modeling the effects of pretreatment conditions on total methane yield during digestion of straw and manure have been compared [52]. The co-digestion of algae, glycerol and waste oil has been studied by determining biochemical methane potential and methane production rate based on a central composite design including a qualitative variable [53]. Modeling of qualitative variables in experimental design is generally performed through dummy variables, where a separate coefficient is used for describing each variable setting. Factorial and fractional factorial designs have been used for kinetic model development based on hydrolysis of lignocellulose for fermentation to ethanol [54]. Experimental design has recently been reported in simulating the effects of solvents for biogas upgrading [55] and distillation of fermented molasses for ethanol production [56] based on central composite designs. Khang et al. [57] presented an interesting approach for evaluating the sensitivity of life cycle assessment (LCA) of biodiesel production from jathropha, waste cooking oil and fish oil. The authors determined important variables for the overall environmental impact of diesel production based on Latin hypercubes, a group of computer generated designs based on random permutation of variable levels. Different thermochemical treatment alternatives of renewable and fossil feedstocks have recently been studied using experimental design. The effects of experimental conditions on oil, char and gas yields from pyrolysis of grass have been determined based on a central composite design [58]. A Box Behnken design and the Taguchi method were used for studying coal gasification for syngas production [59]. The Taguchi methodology was developed in the 1980s and is based on using orthogonal arrays. The designs enable quantifying linear effects and suitable variable combinations for system improvement with an additional emphasis on system or process variability [5].

l

∑ RMSE =

(yi −yi )̂ 2

i=1

l

(20)

where l denotes the number of performed verifications. RMSE is commonly used in multivariate calibration and indicates the average deviation expressed in the original model units. The variance of verification residuals can also be compared with the model residuals based on an F-test [28]. The discussed calculation outputs are illustrated in Table A.2. 4.2. Experimental design in energy applications Experimental design has recently been used for studying the use of alternative fuels in internal combustion engines. Fractional factorial and fractional central composite designs were used for studying 5 emission parameters for replacing diesel with ethanol in a combustion engine [32]. In addition, the effects of experimental conditions on exhaust emissions from replacing diesel with biodiesel and ethanol were determined using a central composite design [33]. Both works also reported the use of optimization functions. Enhancing the performance of diesel engines has been reported using factorial and V-optimal designs [34–36]. In general, V-optimal designs are a specific type of computer generated designs built for minimizing the prediction variance at preset design locations [9]. The effects of experimental conditions on the performance of batteries and fuel cells have recently been determined using experimental design. Important variables for the aging of lithium-ion batteries were determined based on respective p-values or linear models [37,38]. The performance of methanol [39,40] and ethanol [40] fuel cells have been determined based on central composite designs and quadratic models. The effects of temperature, and the relative humidities and stoichiometries of fuel and combustion on the performance of a proton exchange fuel cell were determined using a fractional factorial 25-1 design [41]. Madani et al. [42] used a factorial design for determining the effects of pH and buffer concentration on the performance of a microbial fuel cell by measuring power density, columbic efficiency and chemical oxygen demand change and removal efficiency. The authors found response curvature and added more experiments to form a central composite design, which then enabled the use of a quadratic model. Experimental design has recently been used for studying biodiesel production from a variety of different feedstocks. The effects of 635

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Table 4 Reviewed work on the use of experimental design in energy applications. Application

Variables

Response(s)

Design(s)

N:o exp.

Model

R2/R2adj, R2pre

Ref.

Combustion engine performance

8

5

n.d.

n.d.

n.d.

[32]

Combustion engine performance Combustion engine performance Combustion engine performance Combustion engine performance Lithium-ion battery performance Lithium-ion battery performance Methanol fuel cell performance Methanol and ethanol fuel cell performance Proton exchange fuel cell performance Microbial fuel cell performance Biodiesel production from scum oil Biodiesel production from mixed vegetable oil Biodiesel production from yellow oleander oil Biodiesel production from sun flower oil Biodiesel production from waste cooking oil Catalyst preparation for biodiesel production Biodiesel production from shea butter Biodiesel production from neem oil Biodiesel production from palm kerner oil Pretreatment of straw and manure for biogas Biogas production from algae, glycerol and waste oil Ethanol production from lignocellulose Solvent selection for biogas upgrading

4 9 11 3 4 7 3 3 5 2 4 3 3 3 4 2 4 4 3 2 3

5 6 6 8 1 1 3 1 5 4 1 1 1 1 1 2 2 2 1 1 2

Fractional factorial, fractional central composite Central composite Factorial, V-optimal Factorial, V-optimal V-optimal n.d. Orthogonal array Central composite Central composite 25-1 Factorial, central composite Central composite Box Behnken Fractional central composite 33, Box Behnken Central composite Central composite Central composite Central composite Box Behnken 42, central composite Central composite

31 n.d. n.d. n.d. n.d. 18 17 160 17 20 30 17 15 71 30 11 60 66 17 29 18

0.80–0.99 n.d. n.d. n.d. n.d. n.d. n.d. 0.88–0.95, n.d. n.d. 0.91–0.97, n.d. 0.91, 0.79 1.0, 0.95 1.0, n.d. 0.97–1.0, 0.96–0.99 0.99, 0.98 1.0–0.99, 0.90–0.81 0.96–0.97, 0.85–0.86 0.81–0.98, n.d. 0.99, n.d. 0.94, n.d. 0.98, n.d.

[33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53]

5 4–6

3–5 4–5

25-1, 24 Central composite

48 n.d.

0.76- > 0.90, n.d. 0.96–1.0, 0.69–1.0

[54] [55]

Ethanol production from fermented molasses Sensitivity analysis of biodiesel LCA Pyrolysis of grass for bio-oil Gasification of coal for syngas Pelleting of torrefied biomass Pelleting of torrefied biomass Pelleting of torrefied biomass Hydrothermal treatment of sludge for solid fuel Hydrothermal treatment of sludge for solid fuel Hydrothermal treatment of sludge for solid fuel Ash behavior during hydrothermal treatment of sludge Hydrothermal treatment of sludge for solid fuel Hydrothermal treatment of sludge for solid fuel Hydrotreatment of crude oil Reforming of naphta for gasoline Recycling of purge gas for methanol Steam reforming of methane for hydrogen Pelleting of biomass Pelleting of biomass Pelleting of biomass Pelleting of biomass Pelleting of biomass Sludge drying Sludge drying Sludge drying Economic assessment of a solar power plant Solar heat exchanger performance Micro combustor performance Hydropower system performance Building operational performance Building operational performance Building operational performance Building operational performance

2 5 3 3 2 5 4 3

2 1 3 4 4 11 8 8

Central composite Latin hypercubes Central composite Box Behnken Factorial Custom Custom Custom

11 80 20 13 7 19 29 15

Quadratic n.d. n.d. n.d. n.d. Linear Quadratic Quadratic Interaction Linear, quadratic Quadratic Quadratic Quadratic Interaction Quadratic Quadratic Quadratic Quadratic Quadratic Quadratic Quadratic, interaction Interaction Interaction, quadratic Quadratic Quadratic Quadratic Quadratic Interaction Quadratic Quadratic Linear

1.0, n.d. n.d. 0.98–0.99, 1.0, n.d. 0.74–0.95, 0.78–0.96, 0.83–0.97, 0.86–0.97,

0.55–0.89 0.50–0.91 0.63–0.96 n.d.

[56] [57] [58] [59] [60] [61] [62] [63]

3

6

Box Behnken

15

Quadratic

0.83–0.95, 0.75–0.92

[29]

2

7

Central composite

11

Quadradtic

0.93–0.99, n.d.

[30]

2

3

Box Behnken

15

Quadratic

0.89–0.91, n.d.

[64]

2

9

Central composite

39

Interaction

0.50–0.97, n.d.

[65]

5

1

Orthogonal array

16

Linear

n.d.

[66]

27 16 273 86 11 26 34 27 17 17 19 17 12 31 9 9 25 n.d. n.d. n.d.

Quadratic Quadratic Cubic Quadratic Quadratic Quadratic Quadratic Quadratic Linear, quadratic Quadratic Quadratic Quadratic Quadratic Quadratic n.d. n.d. n.d. Interaction n.d. n.d.

0.99–1.0; 0.98–0.99 0.89, n.d. 1.0, 1.0 0.95, n.d. < 1.0, < 0.98 0.58–0.82, n.d. n.d., 0.27–0.91 0.97, n.d. 0.71–0.90, 0.51–0.79 0.82–0.83, 0.61–0.68 0.86–0.95, n.d. 0.89–0.95, 0.71–0.89 n.d. 0.99–1.0, n.d. n.d. n.d. n.d. n.d. n.d. n.d.

[67] [68] [69] [70] [71] [72] [73] [74] [75] [28] [31] [76] [77] [78] [79] [80] [81] [82] [83] [84]

3 3 4 6 3 3 3 4 4 3 3 3 2 4 4 4 6 10 5 > 100

4 1 2 2 9 3 7 1 4 2 3 2 3 2 1 1 1 16 6 3

3

3 Central composite n.d. Central composite Augmented factorial Augmented factorial Custom Box Behnken Custom mixture Central composite Central composite Central composite Central composite Central composite Orthogonal array Orthogonal array Orthogonal array Fractional factorial Box Behnken Factorial

0.89–0.95

n.d. = not defined.

The effects of experimental conditions on pelleting torrefied biomass have been determined using factorial [60] and customized [61,62] experimental designs. Rudolfsson et al. [61,62] used a design

that combined fractional factorial designs on 4–5 variables for studying the behavior of up to 11 responses. As comparing several response surfaces can be laborious, the authors performed an initial principal 636

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capital and running costs of a residential apartment. The authors used factorial designs on decomposed sub-systems and determined the statistical significance of over a hundred operational variables. The responses were than determined based on the significant variables through multi-objective optimisation. The reviewed work is further summarized in Table 4. As illustrated by the review, experimental design is used in a wide variety of energy applications. Most experimenters reported the use of central composite or Box Behnken designs for modeling or optimization purposes (Table 4). Approximately one fourth of reviewed papers reported using statistically significant model coefficients. It should be noted that a successful statistical interpretation always requires subject knowledge, and including insignificant linear coefficients can have information value as the design was originally based on these variables. However, including insignificant interaction or quadratic model coefficients, or simply using the full quadratic model, can easily lead to overfitting and decreases the degrees of freedom of model residuals often used as an estimate of model error. More than half of reviewed papers reported details on ANOVA for model diagnostics. Only approximately one fifth reported or discussed model residuals. Overall, residual diagnostics are a useful tool for identifying undescribed structure or potential experiments of interest for future work.

component analysis to determine correlations within experimental conditions and measured response variables. This approach was then used also by Mäkelä et al. [63] and Mäkelä and Yoshikawa [30,64] for providing an overview on hydrothermal treatment of industrial sludge for solid fuel applications based on a central composite, a Box Behnken and a custom design including a qualitative variable. The effects of hydrothermal treatment [65] and steam explosion [66] of municipal sludge on respective fuel properties and moisture content have been performed based on a central composite design and an orthogonal array. A Box Behnken design was used for determining the effects and statistical significance of hydrothermal treatment conditions on the fuel properties of recycled paper mill sludge [29]. Experimental design has also been used for optimizing crude oil [67] and low-octane naphta [68] refining through 33 and central composite designs, respectively. The latter work also included a comparison with ANN. Purge gas recycling [69] and methanol reforming to hydrogen [70] have been studied using experimental design. Nobandegani et al. [70] optimized hydrogen production and unreacted methane based on a central composite design with 4 variables. The authors performed a total of 86 experiments in industrial scale. Pelleting and drying processes for biomass have recently been studied using experimental design. Augmented factorial designs were used for determining the effects of experimental conditions on pelleting cassava stem powder [71] and reed canary grass [72] in pilot scale. A custom design composed of three variables on 2–5 levels was also used in pilot scale for determining the effects of moisture content and storage time on the properties of pellets produced from pine sawdust [73]. The energy consumption of pelleting wheat straw according to different pelleting conditions and ultrasonic vibration was determined based on a Box Behnken design [74]. A mixture design was used on different moisture content levels to determine the effects of sawdust blends on industrial pellet properties [75]. Mixture designs are specific experimental designs, where the coded variable levels range from 0 to 1 and the sum of the relative proportions of the mixture ingredients always equals 100%. The use of a pilot scale cyclone for drying different sludge residues from pulp and paper mills has also been studied using central composite designs [28,31,76]. Experimental designs have also widely been used for simulations. The economics of a solar power plant were studied through simulations based a central composite design [77]. The performance and sensitivity of a solar heat exchanger were also determined based on a central composite design [78]. Orthogonal arrays on four factors were used for simulating the performances of a micro-combustor for a thermophotovoltaic system [79] and a hydropower system [80]. Both of these works reported the use of more complex modeling tools than multiple linear regression. An orthogonal array was also used for determining the effects of building materials on the yearly carbon emissions of an office building [81]. The operational performances of buildings were also determined based on fractional factorial [82] and Box Behnken [83] designs. The former work used probability plots for identifying important variable effects, while the latter used ANN and a generic algorithm for modeling and optimisation. Evins et al. [84] reported a case study for optimizing, amongst others, the carbon emissions and

5. Conclusions Experimental design was originally developed for identifying meaningful variation in empirical data within agriculture and biology. The use for optimization came later and was mainly driven by the statistics and chemical societies. This tutorial has summarized and discussed the main aspects of experimental design through a practical example and reviewed recent work within the energy field. Experimental design enables identifying important variables by determining the effects and statistical significance of experimental conditions. The position and character of potential local optima can also easily be determined based on the acquired model. As illustrated by the review, experimental design is used in a wide variety of energy applications through different factorial and optimization designs. Using statistically significant model coefficients helps to avoid overfitting, and model and residual diagnostics are useful in evaluating model performance and identifying potential outlier experiments. As the use of specialized experimental design software will likely increase in the future, familiarity with the linear algebra and statistical concepts behind experimental design will be of help to both the novice and more experienced practitioner. Acknowledgements The author wishes to thank Prof. Paul Geladi from the Swedish University of Agricultural Sciences for discussions on experimental design and data analysis. Bio4Energy, a strategic research environment appointed by the Swedish government, is thankfully acknowledged for supporting this work.

Appendix A See Tables A.1 and A.2.

637

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Table A.1 A central composite design matrix for any two design variables xi and xj. Row n:o

For β0

xi

xj

xij

x i2

x 2j

1 2 3 4 5 6 7 8 9 10 11

1 1 1 1 1 1 1 1 1 1 1

−1 1 −1 1 −α α 0 0 0 0 0

−1 −1 1 1 0 0 −α α 0 0 0

1 −1 −1 1 0 0 0 0 0 0 0

1 1 1 1 α2 α2 0 0 0 0 0

1 1 1 1 0 0 α2 α2 0 0 0

Table A.2 Calculation outputs based on the final model. Experiment

y

ŷ

ei

ei,N

hii

ei,IS

ei,ES

(95% ci)/2a

1 2 3 4 5 6 7 8 9 10 11

2.0 0.85 1.8 1.0 1.7 0.59 1.4 1.2 0.89 1.2 0.94

1.93 0.91 1.84 0.82 1.57 0.55 1.42 1.33 1.06 1.06 1.06

0.07 −0.06 −0.04 0.18 0.13 0.04 −0.02 −0.13 −0.17 0.14 −0.12

0.52 −0.40 −0.31 1.24 0.89 0.25 −0.12 −0.94 −1.23 0.96 −0.87

0.50 0.50 0.50 0.50 0.37 0.37 0.33 0.33 0.20 0.20 0.20

0.73 −0.56 −0.43 1.76 1.12 0.32 −0.14 −1.15 −1.37 1.07 −0.98

0.70 −0.53 −0.41 2.18 1.14 0.30 −0.13 −1.18 −1.48 1.08 −0.97

0.41 0.41 0.41 0.41 0.39 0.39 0.39 0.39 0.37 0.37 0.37

a

Half-fraction of the 95% confidence interval for y .̂

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