A methodology for quantitative comparison of control solutions and its application to HVAC (heating, ventilation and air conditioning) systems

Energy 44 (2012) 117e125

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A methodology for quantitative comparison of control solutions and its application to HVAC (heating, ventilation and air conditioning) systems Karel Macek a, b, *, Karel Marík a a b

Honeywell Laboratories, V Parku 23/26, Prague 148 00, Czech Republic Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13, Prague 120 00, Czech Republic

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 August 2011 Received in revised form 6 February 2012 Accepted 7 February 2012 Available online 4 March 2012

Many control solutions are currently available for controlling energy system operations. Newly developed solutions for established systems are assumed to demonstrate improvement over previous solutions especially in cost savings. We have found this to be true in the area of HVAC (heating, ventilation, and air conditioning) systems. This paper provides a useful methodology for quantitative comparison of control solutions with respect to operational costs, achieved quality of control, and external conditions that could also play important roles.
2012 Elsevier Ltd. All rights reserved.

Keywords: Energy efficiency Data envelopment analysis Local regression Efficiency comparison HVAC systems

1. Introduction Much current research addresses the efficiency of a wide range of energy systems in scheduling [1e3], control [4], and design [5]. Our aim in this article is to introduce a methodology to quantitatively compare two or more solutions for a system that is operated under varying conditions. The paper focuses on scheduling and control solutions, although the solutions could also be applicable to system design. Because, unlike laboratory experiments, large systems never operate under exactly the same conditions, their scheduling and control must be system-specific. These conditions could have a significant impact on achieved outputs and consumed resources, which makes the comparison of two alternative solutions difficult. The requirements for the controlled variables must be satisfied, while the resources must be minimized. This demand makes the comparison even more difficult. It creates a dilemma in having one control solution, c ¼ c1, meets that meets the control requirements, but while using uses

* Corresponding author. Honeywell Laboratories, V Parku 23/26, Prague 148 00, Czech Republic. Tel.: þ420 234 625 937; fax þ420 234 625 900. E-mail addresses: [email protected] (K. Macek), karel.marik@ honeywell.com (K. Marík). 0360-5442/$ e see front matter
2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2012.02.014

more resources, while the other second control solution, c ¼ c2, does not fully meet the requirements fully, even though but appears seems to be more economical in for the resource consumption. The situation is illustrated in Fig. 1. In Fig. 1a, both control solutions work under the same conditions, D, and must fulfill the same requirements for the output which are indeed achieved, i.e., Y ¼ R. Here, the comparison is straightforward, and it is only necessary to choose and apply a twosample statistical test. If the conditions differ for operation of both control solutions, either their actual outputs or their required outputs differ. As shown in Fig. 1b, the comparison presents a complex theoretical challenge, because some performance contracts require the enduser of the control solution to pay the operator a ratio of achieved savings. Using statistics to compare similar, but not exact, situations is nothing new. DEA (Data envelopment analysis) [6] calculates the efficiency of each data point with respect to a set of efficient data points and is still the subject of extensive research [7]. ANOVA (Analysis of variance) and DOE (design of experiment) are two approaches [8] that examine the impact of specific output factors and determine the statistical significance of the difference. Neither approach is suitable for our purposes. DEA assumes the factors have monotonic impact on the outputs, while ANOVA considers only discrete values of factors. The inputs in real systems are frequently

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a

b

Fig. 1. The solutions in a) can easily be compared: only the costs of used resources are calculated and compared. The situations in b) differ in external conditions, outputs, and requirements. In this case, it is difficult to compare c1 and c2.

continuous and not necessarily monotonic. Ambient temperature and its impact on HVAC-related1 costs can serve as a representative example; it is obviously continuous and can be either a positive (during cold seasons) or a negative (hot seasons) factor. This paper extends our presentation [9] and is organized as follows: Section 2 presents the problem formulation. Section 3 discusses the proposed methodology. Section 5 demonstrates its application in the area of HVAC systems. Section 4 extends the basic concept to cover more complex situations. Finally, Section 6 summarizes our conclusions and proposes future research steps and possible advanced applications. The paper also includes Appendix A, describing extensive testing of the methodology with simulations and Appendix B, which provides the basic version of the local regression. 2. Problem formulation The objective of the proposed methodology is to provide engineers, especially in the area of energy efficiency, with a tool to discern a better control solution under changing operating conditions. This comparison is based on the past performance of a single system. The methodology focuses on a relative comparison that is closed to DEA [6]. First, the comparison for a single resource is presented; then, multiple resources are addressed and aggregated in terms of costs. Let past performance be given in the form of a time-series collected over time: (d(i), u(i), c(i), y(i), r(i)) for i ¼ 1; 2; .T. The interpretation is obvious from Fig. 1, where d(i), u(i), y(i), r(i) are vectors, y(i) and r(i), of the same size. The chosen control solution c(i) is a discrete variable, say from {c1, c2}. The problem is to determine the ratio r expressing savings by c2 against c1. We split the data set into two subgroups with respect to c : D1 contains data related to c1

1

HVAC stands for heating, ventilation, air conditioning systems.

and D2 relates to c2. The data is considered to be collected over time. External conditions D, outputs Y, and requests R are called conditions altogether. The data D1 and D2 can be interpreted as a sample from a multivariate random distribution (U, D, Y, R). For both control solutions, we consider the same distribution over (D, Y, R). The same distribution is motivated by the need to average the improvement over typical long-term data, let us say one year. However, the control solutions are assumed to be different for the resources used UjD; Y; R. This approach can be explained with an HVAC example. The ambient conditions and end-user requirements are expected to be more-or-less the same for any applied (and compared) HVAC control strategy. If one control strategy is applied during the winter and another one during the summer, then they cannot be compared. The situation is illustrated in Fig. 2, where the joint distributions are represented by ovals and marginal distributions by distribution functions. The r is defined as the ratio of expected consumption by a new and original solution, marginalized by conditions:


EU ED; Y; R U
D; Y; R; c2 EU Ujc2
r¼ ¼ EU Ujc1 EU ED; Y; R U
D; Y; R; c1

(1)

where E stands for expected value and XjZ for random variable X, conditioned on Z. We do not provide an exact method for calculating the absolute difference between consumptions using particular control strategies. Nevertheless, we assume the elements provided could also be reused for this purpose. 3. Proposed methodology This section describes steps leading from the data to approximations of the r ratio. Because of the complexity of the distribution constructions (D,Y,R) and UjD; Y; R; c, we propose to benchmark based on local regression. However, the first step is appropriate preprocessing.

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output of this analysis is a multi-criteria comparison, e.g., see [12]. Users can decide which control seems to be most suitable for them. Finally, the raw data can be aggregated for some period, say one day. The goal of the aggregation is to reflect the dynamics of the system and aggregate its performance for particular runs. The consumption at a given instant is usually affected by the systems previous operation. A single run can be, for example, one day of HVAC system operation. We use averaging (or median) for d and sum for u. Some aggregation is also applied on r and y, i.e., e(r, y). The aggregation is not limited to averaging and summation; other characteristics of the time-series can also be taken, e.g., for representation of oscillation or trends [13]. Constraints can be aggregated as a percentage of time when they were not satisfied. 3.2. Achieved ratio The first approximation of ratio r from Equation (1) can be calculated as the average consumption of resources for a given control solution: Fig. 2. r is the ratio between expected consumption by the new control solution and the original control solution, marginalized by conditions.

3.1. Data preprocessing

1 X u jD2 j u ˛D i 2 i rA ¼ 1 X u jD1 j u ˛D i i

Data preprocessing is an optional step. If data reliability is a concern, it may be desirable to remove outliers using some accepted method [10]. Other data cleansing techniques are also applicable [11]. Outlier detection and data cleansing must be applied carefully if outliers need to be removed, as they sometimes provide important information for comparison. For example, if the extreme value of an outlier is caused by erroneous control, then considering the efficiency of this control the case should not be ignored. At issue here is that it is not always possible to distinguish outliers caused by unknown factors and outliers caused by inefficient control. Furthermore, detection and removal of outliers should be considered with respect to specific systems. Some specific preprocessing of r and y is recommended. Sometimes, the constraints on the system outputs have to be considered explicitly. For this evaluation, it is possible to assign a utility (or error) e(r, y) value to a given combination of r and y. This value can be taken as another factor of resource consumption, and as such, more representative than r and y. Alternatively, the data can be divided into two or more disjoint parts with different levels of constraint dissatisfaction and the performances can be compared; calculate the r approximation only for similar data. The

(2)

1

where jDi j˛N stands for the number of records in the ith data set. rA denotes the achieved ratio. If the situation is as shown in Fig. 1a, i.e., both control solutions work under the same conditions and have the same control performance, then rA is a suitable approximation of r. 3.3. Reference control baselines In most cases, the impact of external conditions and control performance cannot be omitted; therefore, we introduce the reference control solution baseline, i.e., a mapping u*(r, y, d). Since the reference behavior is usually not available explicitly, it is necessary to use the data to construct it. We considered the following baselines:  Original control - This baseline is constructed from data for one strategy, say D1. In this case the relative savings for the other strategy can be calculated directly:

P

r¼

P ðrðiÞ ;

yðiÞ ;

uðiÞ ˛D

dðiÞ Þ˛D2

uðiÞ

  u* r ðiÞ ; yðiÞ ; dðiÞ 2

Fig. 3. Benchmarking with the original control solution could lead to problems with extrapolation over D2.

(3)

120

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Fig. 4. Benchmarking by ideal control solution might be strongly affected by outliers.

Fig. 5. Average benchmark avoids problems with extrapolation over D2 and is much more robust.

positively or negatively impact the resources consumed. An illustration is provided in Fig. 5. The baseline is calculated using formulas similar to those in the original approach, with the difference that the average-based approach uses both D1 and D2, while the original approach uses D1 only.

However, if the control solutions work under significantly different conditions, the benchmarking model faces problems with extrapolation to the domain of D2, as shown in Fig. 3.  The baseline u* can be constructed via various regression methods, including neural networks or decision trees. We chose to use the well-known local regression, LOESS [14], which has been successfully applied in energy systems and buildings [15]. Refer to B for a basic version of local regression. The methodology is not limited to this approach to regression.  Ideal control - This baseline is constructed from specific data. Only so-called non-dominated records are selected from both D1 and D2. A record (d(i), u(i), c(i), y(i), r(i)) is non-dominated when no other record j would dominate it, i.e., a situation where the consumption was lower u(j) < u(i), external conditions more adverse dðjÞ _dðiÞ , and control error lower e(y(j), r(j)) < e(y(i), r(i)), and while the requirements are the same or even harder r ðjÞ _rðiÞ . This approach is similar to the DEA methodology; however, our early experiments have shown this approach is very sensitive to outliers and requires proper robustness analysis. An illustration of this situation is given in Fig. 4. If all factors are taken into account, the ideal baseline can express the efficiency as such. It might also be hard to distinguish which factors impact the consumption positively and negatively.  Average control - This baseline uses all data and is especially suitable for use when both strategies work under very different conditions. It is not necessary to specify which factors

Local regression is suitable for modeling non-trivial functions; however, it tends to perform extrapolation poorly [16]. Therefore, the average control baseline was chosen, while the ideal and original controls were omitted. For this reason, we did not fit the

Table 1 Results from a numeric example. x

u

c

Ideal u*

Average u*

Original u*

0.494 0.098 0.123 0.058 0.462 0.960 0.963 0.843 0.731 0.594

4.225 3.088 2.815 2.832 4.059 3.410 4.100 3.838 3.037 3.199 1.033

1 1 1 1 1 2 2 2 2 2

2.950 2.815 2.815 2.815 2.939 3.410 4.100 3.220 3.037 2.987

3.917 2.916 2.917 2.914 3.981 3.755 3.756 3.634 3.331 3.570

4.129 2.923 2.931 2.917 4.112 4.180 4.180 4.171 4.163 4.150

1.168 0.883

1.084 0.953

1.225 0.843

rA

rE r ¼ rA =rE

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121

Comparison by ideal solution based benchmarking 4.5 Results for the original solution Results for the new solution Calculated benchmark

Output

4

3.5

3

2.5

0

0.1

0.2

0.3

0.4

0.5 Input

0.6

0.7

0.8

0.9

1

0.8

0.9

1

0.8

0.9

1

Comparison by average solution based benchmarking 4.5

Output

4

3.5

3

2.5

0

0.1

0.2

0.3

0.4

0.5 Input

0.6

0.7

Comparison by original solution based benchmarking 4.5

Output

4

3.5

3

2.5

0

0.1

0.2

0.3

0.4

0.5 Input

0.6

0.7

Fig. 6. Three benchmarking approaches on a small data set provide three obviously different results.

data twice for each control and then put them into Equation (1). Now we can use lazy learning for estimation of consumed resources u, i.e., given the data (d(i), u(i), y(i), r(i)) for i ¼ 1; 2; .T, we obtain u ðiÞ ¼ ðu* ðiÞðdðiÞ; yðiÞ; rðiÞÞÞ. The u*(i) denotes the reference value. Let us consider following ratio:

rE ¼

1 jD2 j 1 jD1 j

X

u ðri ; yi ; di Þ

ðri ; yi ; di Þ˛D2

X

u ðri ; yi ; di Þ

(4)

ðri ; yi ; di Þ˛D1

This number expresses the expected ratio between resources in c2 and c1 days if the baseline control is applied. Using rA and rE, we can estimate the control performance ratio as rzrA =rE . The approximation sign z shows that the ratio is a conservative estimate; if the data for c1 and c2 do not overlap sufficiently, then the performances of control strategies are considered to be similar, i.e. rz1. If the data overlap sufficiently, the estimate will be close to rzrA =rE ; see the left-hand chart in Fig. 5.

3.4. Numerical illustrations The numeric approaches we considered use a very small data set to make the methodology easy to follow step by step. The example is summarized in Table 1. Only one input and one output variable are considered here; their specific values are given in the first two columns of the table. The third column contains indices of the control solution. Now, we can calculate:

1 ð3:410 þ 4:100 þ 3:838 þ 3:0374 þ 3:199Þ rA ¼ 5 ¼ 1:033 1 ð4:225 þ 3:089 þ 2:815 þ 2:832 þ 4:059Þ 5

(5)

The next columns contain the baseline values for each approach. The local regression2 parameter l was determined using leave-oneout minimization for the average and the original baselines:

2

The parameter is introduced in Appendix B.

Fig. 7. The system is fed by gas and electricity, both are paid by money.

laverage ¼ 0.1332, loriginal ¼ 0.1777. The ideal control baseline does not depend on any parameters. From these baseline values, the rE can be calculated and, consequently, so can the r. The baselines are also provided in Fig. 6. Both from the table and the figure, it can be seen that the average-based benchmark justifies these noncomparable results as similar. This numerical example is a simulation used for the extensive testing described in Appendix A. 4. Discussion and further work 4.1. Distinguishing savings based on consumption reduction and savings from dynamic pricing

are low. It is helpful to be able to identify this situation. In the case of a single resource, we could simply divide the overall savings by the resource savings. If more than one resource is consumed, the situation is more difficult. We must introduce an overall resource saving ratio rres. For this ratio, we consider relative costs ai of resources that are proportional to overall costs for particular P r ai ¼ 1. resources and N i

rscheduling ¼

rcosts r ¼ PNcosts r rres ai ri

(6)

i¼1

where Nr is the number of resources. 4.2. Involving long-term changes of the prices

We have already demonstrated how the savings of one resource can be evaluated. In fact, a mix of several resources can be bought, as Fig. 7 shows. For each resource i, we have ri expressing the ratio of expected consumptions for c1 and c2. All the resources are related to some costs. To compare c1 and c2, we can use cost as an input u for previous analyses. We obtain rcosts for costs that correspond to the overall savings. However, the prices of resources can vary throughout the day. The control solution can take this fact into account and operate the system more intensively when the prices

Another aspect of pricing relates to long-term changes. Consider fixed prices that might change while their ratio remains the same. This situation is easy; we simply recalculate costs for older data so the results are comparable. However, the ratio might change, i.e., gas can become relatively cheaper than electricity. In this case, the prices or the ratio can be taken into account as a variable (part of vector d) of the local regression model discussed above, so the information is involved.

Fig. 8. presents daily power and gas consumptions conditioned by average ambient temperature. The black line is the average-based benchmark.

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severe conditions in terms of electricity consumption and but less well under severe conditions in terms of gas consumption.

Table 2 Results from the case study.

Achieved savings Savings by new control

123

1rA 1r

Gas

Electricity

Total

30.93% 26.02%

31.43% 34.26%

31.26% 31.65%

5. Case study: HVAC supervisory control The goal of this study is to compare two supervisory control solutions for controlling HVAC equipment in an administrative building. Typically, the primary goal of any control solution is to sustain a defined comfort level in all building zones, and a secondary objective is to make it cost-effective by minimizing purchased energy costs. We compared control solutions that use the same HVAC equipment and the same building. The solutions differed in their important HVAC set-points (e.g., supply air temperature, supply hot water temperature, supply fan speed, etc.). The first solution, c1, was the original HVAC control with static set-points and few rule-based methods for ambient temperature compensation of heating and cooling water temperature. The second control solution, c2, was a tested, novel control solution using a model-based optimization approach. The control solutions could not run simultaneously, as we had only one controlled system (building). Although the testing site offered a twin-building, we did not adopt a “parallel-run” approach as in Privara et al. (2010), because the occupancy profiles of the twin-buildings were different. Thus, we alternated the control solutions day-by-day basis to allow performance comparison. Two performance measures are associated with an HVAC control solution (i) comfort satisfaction and (ii) operating costs. In our case, the first measure can be omitted as the compared solutions were conservative and did not compromise the defined comfort. Consequently, we considered only the differences in operating conditions. For the HVAC control solutions, the conditions considered were (i) occupancy and (ii) weather. There are three reasons to not consider occupancy: (1) performance is compared just for working days (HVAC is turned off during holidays); (2) the building is large and the expected variance in occupancy is small (especially when taking into account the day-to-day changes of compared control strategies); (3) occupancy information is not available/measured. The most influential weather factor for energy consumption is the ambient temperature. Therefore, the daily costs were conditioned by the average daily temperature, as shown in Fig. 8 and Table 2. The table shows the new control solution performs well under more

6. Conclusion The methodology presented compares two or more control solutions even when operating conditions are dissimilar. The comparison is conservative: for significantly different (non-overlapping) operating conditions, control solutions are rated as equivalent. If the operating conditions overlap, the estimated savings will be similar to savings evaluated by simple averaging. This methodology can be used not only for comparisons, it can also be applied as a driver for automated switching between different control solutions so the best option can be selected for given conditions and requirements. Appendix A. Extensive testing This appendix describes an analytically solvable testing problem and results achieved using the proposed benchmarking approaches and the established data envelopment analysis [6]. Note that we assume the reader is familiar with basic terminology in probability theory and mathematical statistics. Appendix A.1. Testing problem formulation The testing example considers the one-dimensional condition x, uniformly distributed over [0, 1]. For each x the consumed resource Ujx also has a uniform distribution over [mi(x)  si(x), mi(x) þ si(x)] where i ¼ 1, 2 are the control solutions and mi and si are linear functions defined over [0, 1] as follows:

m1 ðxÞ ¼ 3:0 þ 1:5x s1 ðxÞ ¼ 0:5

(A.1)

m2 ðxÞ ¼ 3:5 þ 0:3x s2 ðxÞ ¼ 0:6 þ 0:2x

(A.2)

The sampling from this random function is straightforward for a given control solution i˛f1; 2g; the x is sampled from the uniform distribution. After this sampling, mi(x) and si(x) are calculated and the value of u is sampled. To test whether the control solutions work under different conditions, we used the uniform sampling of x for control solution 1 only the interval [0, 0.6]; for control solution 2, we sampled interval [0.4, 1]. Fig. A.9 shows these two solutions, including some samples on restricted intervals. It is difficult to distinguish which of them is more efficient in terms of Equation (1).

Fig. A.9. Shows the formulated testing problem.

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Appendix A.2. Analytical solution Even though error distribution is usually assumed to be normally distributed, we used the uniform distributions because the r from Equation (1) can be calculated analytically. The r is actually a division of two mean values we call them expected consumption:

Z i ðxÞ Z1 mi ðxÞþs EU Ujs¼i ¼

u 0 mi ðxÞsi ðxÞ

Z1 " ¼ 0

Z1 ¼

1 dudx 2si ðxÞ

(A.3)

# 2 mi ðxÞþsi ðxÞ

ðmi ðxÞ þ si ðxÞÞ ðmi ðxÞ  si ðxÞÞ 2$2si ðxÞ 2

dx

Thus, for maximally efficient records xi ¼ 1, the efficiency expresses how much a particular record is able to use the resources for the effort, e.g., x ¼ 0:25 means that a particular record uses x1 ¼ 4 times more resources than needed for given effort. To attain some similar ratio for p, we use the ratio of the average inverted DEA efficiencies:

1 X 1 x jD2 j i˛D i 2 rx ¼ 1 X 1 jD1 j

(A.4)

(A.8)

xi

i˛D1

mi ðxÞsi ðxÞ

mðxÞdx

(A.5)

Appendix A.4. Comparison of benchmarking approaches

0

and considering the linear form of mi ðxÞ ¼ ai x þ bi we obtain:

Z1

mi ðxÞdx ¼

0

Z1

ai x þ bi dx ¼

ai 2

x þ bi

(A.6)

0

Thus,

EU Ujs¼1 ¼ 3:75

and

EU Ujs¼2 ¼ 3:65.

Consequently

r ¼ 3:65=3:75 ¼ 0:9733. Benchmarking methods should be able

For more extensive testing, we performed 1000 simulations. The simulations started with data sampling from the described distribution, including five points for each control solution. Benchmarking was then performed as described in 3.4, and the calculated values rE were stored. After the simulations were performed, the results were assessed. They are shown in Fig. A.10 as histograms. Table A.3 offers some statistics, including results of two statistical tests:

to closely approximate this value. Appendix A.3. DEA-based approach For a given data set, D, DEA is able to calculate relative efficiencies xi for all records where conditions x will be treated as outputs and consumptions and u as inputs, because DEA assesses the ratio of achieved effort and given resources. For multidimensional cases, linear programming is used to calculate efficiencies. However, for a single-input, single-output case, the efficiency is calculated as follows:

xi ¼

xðiÞ uðiÞ maxj˛D

(A.7)

xðjÞ uðjÞ

 Test 1 is a two-sided sign test for whether the median is the analytically known value 0.9733. For this test to pass, the null hypothesis must be accepted.  Test 2 is a one-sided sign test for whether the median is 1 with an alternative hypothesis that the value is lower than 1. This test is passed if the null hypothesis is rejected. The sign tests were used because the assumption of normality is obviously not relevant, especially for the DEA-based approach. Both tests were performed with a ¼ 0.05. The results show that only the ideal-based benchmark satisfies the requirements. On the other hand, we note that (i) the sample size is very small and (ii) the error had uniform distribution. If the samples were larger and the error normally distributed, the average-based approach might provide better results.

ideal

200 100 0

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

average

200 100 0

original

100 50 0

dea

40 20 0

Estimated ρ

Fig. A.10. Histograms of estimated r values for particular benchmarking approaches. The red line points out the true value of r ¼ 0.9733. DEA is obviously irrelevant.

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References

Table A.3 Statistics and test on r estimates for particular approaches.

Average Stdev Median Test 1 Test 2

Ideal

Average

Original

DEA

0.976 0.061 0.974 Accepted Rejected

1.018 0.044 1.016 Rejected Accepted

1.020 0.096 1.015 Rejected Accepted

2.465 0.860 2.282 Rejected Accepted

Appendix B. Local regression In this Appendix, the basic version of the local regression is Nd provided. Let us consider a data set D ¼ ðuðiÞ ; xðiÞ Þi¼1 where xðiÞ ¼ Nx ðiÞ ðiÞ ðiÞ ðr ; y ; d Þ˛R represents all considered conditions. The baseline u* is calculated for a given query point x as follows:

0 Ki ¼ exp@ 

Nx  X

xj 

ðiÞ xj

. 2

lj

1 A

i ¼ 1; 2; .; Nd

(B.1)

j

K li ¼ PN i d

j¼1

u* ðxÞ ¼

Kj

Nd X

i ¼ 1; 2; .; Nd

li $ui

125

(B.2)

(B.3)

i¼1

where lj > 0 are parameters of the method called bandwidths. Their optimal value can be set up using minimization of the leaveone-out loss function Wasserman (16, p. 70).

[1] Marík K, Schindler Z, Stluka P. Decision support tool for advanced energy management. Energy 2008;33:858e73. [2] Moghaddam A, Seifi A, Niknam T, Pahlavani M. Multi-objective operation management of a renewable mg (micro-grid) with back-up micro-turbine/fuel cell/battery hybrid power source. Energy 2011;36(11):6490e507. [3] Tina G, Passarello G. Short-term scheduling of industrial cogeneration systems for annual revenue maximisation. Energy 2012;42(1):46e56.  [4] Prívara S, Siroký J, Ferkl L, Cigler J. Model predictive control of a building heating system: the first experience. Energy and Buildings 2010;43:564e72. [5] Cheng W, Mei B, Liu Y, Huang Y, Yuan X. A novel household refrigerator with shape-stabilized pcm (phase change material) heat storage condensers: an experimental investigation. Energy 2011;36:5797e804. [6] Charnes A, Cooper W, Rhodes E. Measuring the efficiency of decision making units. European Journal of Operational Research 1978;2:429e44. [7] Chen W, McGinnis L. Reconciling ratio analysis and DEA as performance assessment tools. European Journal of Operational Research 2007;178(1):277e91. [8] Bailey R. Design of comparative experiments. Cambridge series on statistical and probabilistic mathematics. Cambridge, United Kingdom: Cambridge University Press; 2008. [9] Macek K, Marík K. A methodology for quantitative comparison of control solutions with respect to costs, quality and external conditions. Chemical Engineering Transactions 2011;25:459e64. [10] Papadimitriou S, Kitagawa H, Gibbons P, Faloutsos C. LOCI: fast outlier detection using the local correlation integral. In: Proceedings of 19th international conference on data engineering. Bangalore (India): 2003. pp. 315e326. [11] Hernández M, Stolfo S. Real-world data is dirty: data cleansing and the merge/ purge problem. Data Mining and Knowledge Discovery 1998;2:9e37. [12] Triantaphyllou E, Evans G. Multi-criteria decision making in industrial engineering. Computers and Industrial Engineering 1999;37(3):505e6. [13] Kantz H, Schreiber T. Nonlinear time series analysis. Cambridge nonlinear science series. Cambridge, United Kingdom: Cambridge University Press; 2004. [14] Cleveland WS. Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association 1979;74:829e36. [15] Stluka P, Marík K, Rojí cek J. The role of predictive models in energy efficiency optimisation of industrial plants and buildings. Chemical Engineering Transactions 2009;18:725e30. [16] Wasserman L. All of nonparametric statistics (Springer texts in statistics). Secaucus. New Jersey, USA: Springer-Verlag New York, Inc; 2006.