Applied Energy 191 (2017) 492–501

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Applied Energy
journal homepage: www.elsevier.com/locate/apenergy

Power capacity profile estimation for building heating and cooling in

demand-side management
Juan A. Gomez ⇑, Miguel F. Anjos
GERAD and Department of Mathematics and Industrial Engineering, Polytechnique Montreal, C.P. 6079, Succ. Centre-Ville, Montreal, QC H3C 3A7, Canada

h i g h l i g h t s

We present a new methodology to estimate power capacity profiles.

We use a classification approach to estimate the capacity.

Our methodology works with an existing demand-side management module.

We take advantage of the structure of the problem.

We report the performance of our approach on a real-world-based scenario.

a r t i c l e i n f o a b s t r a c t

Article history: This paper presents a new methodology for the estimation of power capacity profiles for smart buildings.
Received 29 September 2016 The capacity profile can be used within a demand-side management system in order to guide the building
Received in revised form 21 January 2017 temperature operation. It provides a trade-off between the quality of service perceived by the end user
Accepted 27 January 2017
and the requirements from the grid in a demand-response context. We use a data-fitting approach and
a multiclass classifier to compute the required profile to run a set of electric heating and cooling units
via an admission control module. Simulation results validate the performance of the proposed method-
Keywords:
ology under various conditions, and we compare our approach with neural networks in a real-world-
Smart buildings
Power demand
based scenario.
Residential load sector Ó 2017 Elsevier Ltd. All rights reserved.
Least squares
Parameter estimation
Classification

1. Introduction baseboards account for 27% of heating equipment, reaching 66%

in the province of Quebec. On the other hand, the province of
In the context of power systems, reducing peaks and the fluctu- Ontario is typically a summer-peaking region due to the high tem-
ation of consumption brings stability to the system and benefits to peratures during that season and the high penetration of air-
the players in the power supply network. In this respect, demand- conditioning systems [3,4].
response (DR) programs and demand-side management (DSM) Several authors have published DSM-related results. Normally
systems encourage and facilitate the participation of the end users their research motivation is oriented to load management, user
in the grid decisions. This participation is increasing with the behavior, cost performance, and curve shaping. Imposing a capac-
development and implementation of smart buildings. DR programs ity constraint is a common idea among these approaches. Costanzo
have mostly been oriented to large consumers, but smart buildings et al. [5] propose a multilayer architecture that provides a scheme
can exploit the DR potential in residential and commercial build- for online operation and load control given a maximum consump-
ings as well. These represent around 70% of the total energy tion level. In the stochastic DSM program in [6], a DR aggregator
demand in the United States [1]. In Canada, space heating is imposes a capacity constraint. Bidding curves and price analyses
responsible for more than 60% of the total residential energy con- are reported in order to guide end-users about increasing capacity.
sumption, due to the cold climate [2]. Across the country, electric Rahim et al. [7] evaluate the performance of several heuristic-
based controllers. They define the load management as a knapsack
problem with preset power capacities for each time slot. In a
⇑ Corresponding author. similar way, [8] assumes a consumption limit that allows the
E-mail address: juan.gomez@polymtl.ca (J.A. Gomez).

http://dx.doi.org/10.1016/j.apenergy.2017.01.064
0306-2619/Ó 2017 Elsevier Ltd. All rights reserved.
 J.A. Gomez, M.F. Anjos / Applied Energy 191 (2017) 492–501 493

Notation

h 2 f1; 2; . . . ; Hg set of time frames in horizon QoSh quality of service in time frame h
t 2 f1; 2; . . . ; Sg set of time steps in time frame h (same for every h) d
QoS quality of service of the prediction model in time frame
h
i 2 f1; 2; . . . ; Ig set of loads h
Nh number of requests received in time frame h T temperature (°C)
Pi power level of load i (kW) T eh external temperature in time frame h (°C)
Ch power capacity in time frame h (kW) P power levels of the loads in each scenario


1 if a request is created by load i in time step t X discrete set of capacities
r i;t


0 otherwise x2X capacity class
1 if request from i is accepted in time step t
xi;t
0 otherwise

activation of only one load at a time. Li et al. [9] look for an optimal This paper proposes an approach for the estimation of a power
allocation of capacities based on a queueing strategy. The service capacity profile that works in combination with the admission con-
provider determines the capacity to assign to each user from a troller (AC) module presented in [5]. This profile is used to ensure
set of renewable resources. enough power to meet the demand for the next planning horizon
The idea of capacity subscription is explored in [10], where the (e.g., the next day in a day-ahead DR market). This novel approach
individual consumer’s demand is limited in a competitive market. takes advantage of the structure derived from the estimation prob-
On the other hand, the heuristic algorithm proposed in [11] aims to lem to compute capacity profiles efficiently and reliably. Estimat-
minimize the error between the actual power curve and the objec- ing the capacity that will be necessary allows us to define a
tive load curve by moving the shiftable loads. In this case the relationship between the total expected demand and the level of
objective load curve can be seen as a soft constraint capacity service the user desires while providing DR. In this scenario the
profile. user will book a variable maximum power capacity per time frame
A variation of the capacity limit is presented in [12], where each over the planning horizon, ensuring a pre-established level of ser-
individual user has a predefined budget to maximize his/her vice. This approach could also include external factors such as peak
satisfaction. control and pricing policies. The motivation is that a defined power
All the approaches mentioned represent the capacity as a given budget limits the consumption and encourages load shifting. It also
parameter, and some of them recognize the importance of using a facilitates the integration of differential pricing for both energy and
forecasting tool to determine its value. Estimating the user con- power.
sumption is a key step in the decision-making process for users This paper is structured as follows. We describe the proposed
and for higher levels in the power system. Relevant publications methodology in Section 2. We give simulation results for the
can be found in the load-forecasting literature. Suganthi and real-world-based scenario in Section 3, and Section 4 presents
Samuel [13] give a comprehensive review of forecasting methods our conclusions.
from classical time series to more sophisticated machine learning
tools. 2. Power capacity profile
Load estimation methods are classified depending on the level
of aggregation of the input data: they can be bottom-up or top- Fig. 1 shows the application of the AC module presented in [5].
down [14]. Bottom-up models extrapolate the behavior of a larger The online algorithm in the AC has four stages. First, the space
system based on its inner elements. Top-down models make deci-
sions from a global perspective and share them among all the
subsystems.
Within these two categories different approaches have been
used to estimate the energy demand. Ahmed et al. [15] compare
artificial neural networks and the auto regressive integrated mov-
ing average, showing the effect on the scheduling of storage
devices. Jain et al. [16] use support vector regression to evaluate
the impact of the time and space granularity inside a multi-
family unit. Al-Wakeel et al. [17] use a k-means-based load estima-
tion method to compute future load profiles using complete and
incomplete past information.
Logistic and Poisson regression are used in [18] to estimate
energy demand in a large aggregated population. In a similar
way, [19] presents a short-term forecasting method for aggregated
loads, specifically in buildings with daily or seasonal patterns of
consumption. Mohajeryami et al. [20] present an error analysis
for different load estimation methods that are used in real-world
operations. They highlight the importance of an accurate estima-
tion for exploiting the DR potential.
On the other hand, when the prediction output belongs to a
discrete set of categories the estimation can be defined as a
classification problem. Some related energy problems are treated
in this way: price forecasting in [21] and wind power ramp events
in [22]. Fig. 1. Admission controller.
494 J.A. Gomez, M.F. Anjos / Applied Energy 191 (2017) 492–501

heaters and the air conditioners create requests r i;t when the room of service is desired. This decision-making by the user is especially
temperature is out (or going out) of the thermal comfort zone. Sec- important under time-of-use pricing conditions because the cus-
ond, the algorithm sorts all the requests from the highest to the tomer can profit from the cheaper time frames by reshaping the
lowest priority value; the priority value is the normalized differ- load curve while ensuring the desired QoS.
ence between the temperature in the room and the external tem- In a smart building it is possible to compute the QoS from the
perature. Third, the AC accepts the highest priority requests until information provided by thermostats and smart loads connected
the given capacity C h is consumed; the other requests are rejected. to the AC. In the spirit of [23], we define the QoS for each time
Finally, it sends the signal xi;t back to each smart load i either to run frame h as follows:
(if accepted) or to stand by for the next time step (if rejected). 8 PI PS
< i¼1 t¼1 xi;t
Fig. 2 presents a basic example of the AC operation. A smart
QoSh ¼ Nh
 100% Nh > 0 ð1Þ
house with two rooms, R1 and R2, is simulated over a horizon of :
100% Nh ¼ 0;
5 time frames. Each time frame has 10 time steps where the smart
loads can send requests. Typically, a time frame would be equiva- P P
where N h ¼ Ii¼1 St¼1 r i;t .
lent to an hour in a realistic scenario. There is a 1.5 kW space hea- The accepted requests have to satisfy
ter in each room, and the external temperature is 5 °C (Fig. 2(a)).
We can see the peak reduction obtained by the AC in Fig. 2(b); X
I
xi;t Pi 6 C h 8t 2 f1; 2; . . . ; Sg: ð2Þ
the end-user agrees to have a preset power capacity (dashed red
i¼1
line), which constrains the consumption to at most 1.5 kW. The
peak of consumption, for this example 3 kW, would be attained Eq. (2) indicates that the AC accepts requests until the capacity limit
when the two space heaters are being used at the same time step. is reached. In the framework of this article we assume that both air-
Fig. 2(c) and (d) show the internal temperature in each room conditioning units and electric baseboard heaters have a constant
within a certain comfort zone. In a similar way, we can see the time level of consumption [24]. Let C h 2 X, where X is a set of capacities
steps where the heaters are working in Fig. 2(e) and (f). For more that can work in combination with the AC and the set of loads. In
details about the AC algorithm and the heat transfer equations other words, we do not want a capacity to operate a fractional num-
we refer the reader to [5]. ber of loads in the time step t. Given that X is a discrete set we can
In the previous example the capacity profile suffices to keep the define the classification problem
average internal temperatures (21.8,22.2) °C in the comfort zone UðT eh ; QoSh Þ ¼ C h ð3Þ
[20–24] °C. In the event that the temperature in a room goes out
of the comfort zone during a time step, the space heater will that determines C h 2 X for a given external temperature T ehand the
increase its priority value, and the AC will accept the request in QoSh defined by the user. We solve this classification problem using
the next time step. The capacity profile also achieves a peak shav- a three-step approach: selection of the training set from historical
ing effect. However, alternating the use of the heaters might not be data, function fitting, and final classification. We illustrate the steps
enough to ensure a comfortable internal temperature if the exter- in this section with a group of space heaters; Section 3 includes
nal temperature is extremely low; a higher capacity profile might experimental results for both types of loads.
be required. This decision becomes more complex if we increase
the number of space heaters and if they have different power 2.1. Sampling from historical data
requirements.
We introduce the quality of service (QoS) index to quantify the The real data is obtained from the smart energy management
impact of a given capacity on the whole system. The general idea of system, which records the accepted requests, the rejections, and
QoS is that the user should be willing to pay more if a higher level the evolution of the QoS over time. We simulate this historical data

6 3
Capacity
External Temperature
P(kW)

2
T(°C)

Consumption
5
1
4 0
1 2 3 4 5 1 2 3 4 5
h h
(a) (b)
25 25
Temperature R1 Temperature R2
T(°C)

T(°C)

20 20
1 2 3 4 5 1 2 3 4 5
h h
(c) (d)

Activation Heater in R1 Activation Heater in R2

1 2 3 4 5 1 2 3 4 5
h h
(e) (e)
Fig. 2. Example of results from admission controller.
 J.A. Gomez, M.F. Anjos / Applied Energy 191 (2017) 492–501 495

to understand the system dynamics and to implement a prediction 100

model. The simulation conditions are:
90

The set of heaters is composed of four identical units of 1.5 kW 80
of consumption.

The heat transfer is computed using the specific heat and Four- 70
ier’s law formulations implemented in [5] (see Section 3 for 60

QoS (%)
more details).

The external temperature corresponds to the complete year 50
2013 (8760 hours) in the Montreal area [25].
40

The comfort intervals for the internal temperatures are taken
from the ISO 7730 standard analyzed in [26]. For an office cate- 30
gory B the intervals are [20–24] °C and [23–26] °C for heating
and cooling respectively. 20 C=1.5 kW
C=3.0 kW

C h is randomly chosen from X ¼ ½1:5; 3:0; 4:5; 6:0 based on the 10 C=4.5 kW
interval of temperature; the highest capacities are not neces- C=6.0 kW
sary during the warmer days (for example, with T eh ¼ 19  C 0
-20 -15 -10 -5 0 5 10 15 20 25
every value in X will return a QoSh near 100%, affecting the
Temperature (°C)
quality of the data training set and the estimation).
Fig. 4. Graph of QoS vs. temperature for the sampled historical data.
Fig. 3 shows the frequency of the external temperature intervals
in the historical data; this is clearly an imbalanced set. This imbal-
ance is generated by the similar weather in Spring and Fall. The dh ¼ b1
QoS e þ b3 C h þ b4 ; ð4Þ
temperatures between 0 and 20 C would have a significantly 1 þ eb2 T h
higher weight in a fitting process. We use random under-
d h is the quality of service from the prediction model at
where QoS
sampling [27] in order to match the number of points in the minor-
ity group from the temperatures below the comfort interval. time frame h.
Fig. 4 shows the hourly QoS results for the balanced set. We can Additionally, we will compare two different optimality criteria:
identify several characteristics of the system behavior: the least squares value (LSV) and the least absolute value (LAV).
Typically, the LSV gives more weight to distant points while the

As the temperature increases the QoS converges to higher val- LAV is resistant to outliers [28].
ues; with fewer requests the selection of a capacity level is a The optimization problems are:
less sensitive issue.
X
H 2

The selection of the capacity level has a big impact on the QoS in min d hÞ
ðQoSh  QoS ð5Þ
b1 ;b2 ;b3 ;b4
lower temperature conditions. h¼1

The QoS seems to behave asymptotically for higher and lower
temperatures. X
H
min d hj
jQoSh  QoS ð6Þ
b1 ;b2 ;b3 ;b4
2.2. Data fitting h¼1

Fig. 5 shows the results for a least-squares fitting of a sigmoid

Once we have identified these features in the data set we can function.
solve an optimization problem for the capacity estimation. We fit Once we have solved the optimization problem (5) or (6) we can
the sigmoid function use (4) to compute the expected required capacity for the desired
QoS.

100
1800
90
1600
80
1400
70
1200
60
Frequency

QoS(%)

1000
50
800
40
600
30
400
20 C=1.5 kW
C=3.0 kW
200 10 C=4.5 kW
C=6.0 kW
0 0
-30 -20 -10 0 10 20 30 40 -20 -15 -10 -5 0 5 10 15 20 25
Temperature (°C) Temperature(°C)

Fig. 3. Histogram of hourly external temperatures in Montreal, Canada for 2013. Fig. 5. Fitted sigmoid function.
496 J.A. Gomez, M.F. Anjos / Applied Energy 191 (2017) 492–501

2.3. Motivation for using a sigmoid function 2.4. Classification

The selection of a sigmoid function has both a graphical justifi- As stated previously, we have a discrete set of capacities that
cation and an interesting background. We provide intuition into are suitable for the performance of the system. We solve for C h
why it works for the heating case; the cooling case is similar. This in (4) in order to compute the continuous signal C ch . Finally, we
analysis applies to any external temperature regardless of the time use the multiclass classifier
frame where it occurs; therefore we omit the subscript h and use N
in the place of N h to increase readability. ch  xj
C h ¼ arg minj C ð13Þ
x2X
We make the following assumptions:
to find the required capacity.
0
b e then NðT e0 Þ > Nð T

If T e < T b e Þ for any temperatures T e0 and T
be. Fig. 6 shows the effect of the classifier; it assigns areas to each of

C 2 ½C min ; 1Þ where C min ¼ maxðPi Þ. the capacities based on the midpoints for each pair of sigmoid

Each load generates at most one request per time step, and curves from Fig. 5.
therefore the maximum number of requests per step equals I.

There exists a temperature T e e at which all the heaters generate
3. Experimental results
e e Þ ¼ I  S.
requests at every time step, and therefore Nð T
In the previous section we introduced the methodology with an
Considering the worst-case scenario for any time frame in Eqs. example for a given set of homogeneous space heaters. In this sec-
(1) and (2), we have: tion we carry out several experiments to assess and validate the
PI PS performance of the proposed methodology under different
t¼1 xi;t conditions.
QoSð Te e ; C min Þ ¼ i¼1
ð7Þ
IS It is important to ensure coherence in the thermal system when
defining the set of loads. The loads must keep the temperature in
X
I the comfort range during the warmest and coldest time frames
xi;t Pi 6 C min 8t 2 f1; 2; . . . ; Sg ð8Þ in the data sets. This design step must include the specific features
i¼1 of the building such as size, surfaces in contact with external tem-
Eq. (8) allows us to accept at least one request at every time peratures, wall insulation materials, and thermal load inside the
step. Therefore, the total number of accepted requests satisfies: room. A poorly balanced thermal system could lead to a QoS of
100% with temperatures far from the comfort zone.
X
S X
I At the end of each time step, we compute the temperature in
xi;t P S: ð9Þ the rooms using the same thermal equations as in [5]:
t¼1 i¼1
tot
dQ
room ¼ mroom C room ; ð14Þ
After substituting (9) into (7) we can obtain a minimum QoS:
dT
1
QoSð Te e ; C min Þ P ð10Þ exch
I dQ A
¼ K wall ðT e  T room Þ; ð15Þ
We can see similar behavior for scenarios with temperature dt v
_e
T >T e e and NðT_e Þ < Nð T
e e Þ. Let F be the minimum number of time
steps where requests are received. Since each load i will request Q tot ¼ Q exch þ gPi ; ð16Þ
at most once per time step, we have:
where K wall ¼ 4:8  104 kW/m°C is the average thermal conduc-
& ’
tivity of the wall, and g ¼ 100% is the efficiency of the loads. We
NðT_e Þ NðT Þ_e
F¼ ¼ þ a; 1 > a P 0: ð11Þ choose a room size of 60 m3, which corresponds to an air mass of
I I
mroom ¼ 72 kg with a specific heat capacity C room ¼ 1:0 kJ/kg°C.
The variable F also becomes the minimum number of accepted The surface area in contact with the external temperature is
requests due to the C min in Eq. (8). By substituting (11) into (1) we
obtain:
100
PI PS C=6.0 kW
t¼1 xi;t F 90
QoSðT_e ; C min Þ ¼ i¼1
P : ð12Þ
NðT Þ_e ðF  aÞI 80
C=4.5 kW
When a ¼ 0 we get the same condition as in Eq. (10). 70
A sigmoid function helps to represent the asymptotic extremes 60
QoS(%)

and monotonic behavior of the QoS. In the first case, we see how C=3.0 kW
50
the QoS is bounded below in Eqs. (10) and (12), and it is bounded
above by definition ðQoS 6 100Þ. In the second case, the tempera- 40 C=1.5 kW
0
ture and requests are inversely proportional (if T e < T b e then
30
0
NðT e Þ > Nð T b e Þ), so QoSðT e Þ is monotonically increasing. Using a lin-
20
ear function would capture the monotonic condition but not the
asymptotic extremes. 10
For cooling systems we would change the first assumption to
0
b e , giving NðT e0 Þ > Nð T b e Þ. This leads to a similar monotoni- -20 -10 0 10 20
Te > T Temperature(°C)
cally decreasing sigmoid function over the interval of external tem-
perature where cooling is required. Fig. 6. Classification areas.
 J.A. Gomez, M.F. Anjos / Applied Energy 191 (2017) 492–501 497

A ¼ 12 m2 with a thickness of v ¼ 0:2 m. This remains constant for and heterogeneous loads are tested with a X set that was defined
all the experiments. separately from the loads. Finally, scenarios 7 and 14 contain a
For a more realistic scenario both types of loads are managed by heterogeneous set of loads and all possible combinations in X.
the AC; the space heaters and the air conditioners will create In general, we observe a better performance in the sigmoid fit-
requests when the temperature in each room is moving out of ting (SLAV and SLSV) than in the polynomial cases (PLAV and
the comfort zone. PLSV). There is no clear difference in terms of the fitting criterion.
The experiments include: The sigmoid function seems to be competitive with both NNs in the
first six scenarios of each table.

Sets P with homogeneous and heterogeneous power P i values. As stated before, the sigmoid function provides a better repre-

Three different types of X sets: computed from all possible com- sentation of the structure of the problem. Fig. 7 shows the classifi-
binations of values in P; computed from some of the combina- cation areas obtained by fitting the sigmoid and polynomial
tions in P; and given by an external entity. functions for scenario 2. For a QoS of 90%, we see that the polyno-

Two fitted functions. mial function gives a transition between areas either before or

Two optimality criteria: LAV and LSV. after the sigmoid function. If it is before, T 2 ð18; 8Þ  C, we will

Comparison with two neural networks (NNs) with different obtain a worse QoS and lower temperatures in the rooms. If is after,
topologies. T 2 ð2; 8Þ  C, we will have extra capacity that is not required. This
lower utilization of the capacity becomes more important if the
The experiments are carried out in two stages. In the training user is paying in advance for a resource that will not be used.
stage we reproduce the approach presented in Section 2 in order On the other hand, scenarios 7 and 14 are significantly differ-
to determine the classification areas. Then in the test stage we ent: the NNs have considerably better performance than any other
use the classification areas to estimate the capacity profiles for approach. Looking deeper into the characteristics of these scenar-
given levels of the QoS. When the profiles have been computed, ios we see a special condition: several values in X can generate
we run a simulation to verify the actual QoS performance. the same QoS at the same temperature. We may have the same
In Sections 3.1 and 3.2 we illustrate the methodology on a performance in scenario 7 for x ¼ 4 and x ¼ 4:5 if the three hea-
three-load instance: an apartment with three rooms. In Section 3.3 ters send requests at the same time. In the first case, the AC will
we report results for an instance with 50 loads to demonstrate the accept P1 and P3 and leave P2 for the next time step. In the second
scalability of our methodology. case the order of acceptance changes but the QoS is the same. Fig. 8
shows the training set for this scenario; we can see how C ¼ 4 is
3.1. Training for three-load instance distributed over its adjacent classes.
Although the NNs have a better training performance, they
We required two training sets: one for heaters and one for air might minimize the confusion value by eliminating one of the
conditioners. Each training set is defined over the corresponding classes. Let W x be the set of points that belong to class x, and
interval of temperature (T eh 6 20  C for heaters and T eh P 26  C for let W 1x and W 0x be the subsets of points correctly and incorrectly
air conditioners) and randomly chosen as in Section 2.1. The histor- classified respectively. Let C be the total number of misclassified
ical sets are simulated using the hourly temperature in Montreal points. The approach presented in this article separates any two
for the year 2013 (8760 data points). contiguous sets following the fitted function, and therefore
As mentioned before, we will compare this methodology with W 11 þ W 01 ¼ jW 1 j; W 12 þ W 02 ¼ jW 2 j, and W 01 þ W 02 ¼ C.
two other approaches. In the first case, we use the polynomial If we assume that the NN eliminates class 2 we have
function  . We can conclude
W 11 ¼ jW 1 j; W 02 ¼ jW 2 j, and W 12 þ W 02 ¼ jW 2 j ¼ C
that eliminating one class improves the confusion (i.e., C  < C) if
d h ¼ b þ b Ch þ b T e þ b T e Ch
QoS ð17Þ
1 2 3 h 4 h
W 12 < W 01 .
in the fitting step. A priori the sigmoid function gives a better rep- At this point we can see the advantage of exploiting the features
resentation of the historical set due to its monotonically increasing of the problem. In the approach presented in this paper the fitted
behavior and the asymptotic extremes. The function in Eq. (17) cap- function acts as a constraint that represents the structure of the
tures only the monotonic condition. To fit each function we solve a data sets. On the other hand, the flexibility of the NNs allows a
nonlinear optimization problem using the BFGS method; it finds a lower misclassification, but we see in Section 3.2 that this has an
solution in a few seconds. unexpected impact on the QoS.
We use NNs, which are widely used in many different types of
problems, as a second benchmark. For classification problems the 3.2. Results for three-load instance
NN typically has the same number of neurons in the output layer
as the number of classes. The NN computes the probability that The experiments use data for a period of two years (2014 and
each input belongs to each class, and it chooses the class with max- 2015) for the Montreal area (17520 data points). The user sets a
imum probability. We implemented two NNs with A ¼ 1 and B ¼ 2 QoS of 90%. Figs. 9–14 show the results for scenarios 2 and 7 (heat-
hidden layers (5 neurons each), cross entropy as a performance ing) and scenario 14 (cooling). These box plots contain the mini-
measure in the learning process, and a validation subset of 30% mum value, maximum value, and interquartile range for the
of the points. The training time of the NNs varies between 10 hourly QoS and the hourly average temperature in the three rooms
and 20 s using scaled conjugate gradient backpropagation. for each of the methods compared.
Finally, the total confusion or missclassification index measures For scenario 2 (Figs. 9 and 10) we see that the sigmoid and NN
the performance of each approach. It indicates the percentage of cases perform slightly better than the polynomial function.
the total set of data that was incorrectly classified. Although the QoS and the temperature do not vary significantly,
Tables 1 and 2 show the training results for the different scenar- the use of the resource differs: the polynomial function reports
ios and approaches. Scenarios 1–7 and 8–14 correspond to heating around 60% of utilization of capacity while the other four methods
and cooling respectively. In scenarios 1–3 and 8–10 the loads are achieve a utilization between 70% and 75%. This effect was previ-
homogeneous and the X set corresponds to all possible combina- ously observed in Fig. 7. Scenarios 1, 3 to 6, and 8 to 13 have similar
tions of the loads. In scenarios 4–6 and 11–13, both homogeneous results.
498 J.A. Gomez, M.F. Anjos / Applied Energy 191 (2017) 492–501

Table 1
Confusion (%) in training stage for the heating scenarios.

Scenario P X PLAV PLSV SLAV SLSQ NNA NNB

1 ½1:5; 1:5; 1:5 ½1:5; 3:0; 4:5 28.31 33.92 12.54 13.12 11.25 10.15
2 ½2:0; 2:0; 2:0 ½2:0; 4:0; 6:0 18.94 20.38 13.48 11.70 15.76 10.10
3 ½2:5; 2:5; 2:5 ½2:5; 5:0; 7:5 20.58 25.00 20.92 17.95 15.70 10.88
4 ½1:5; 1:5; 1:5 ½2:5; 4:0; 6:0 32.04 28.50 10.01 14.47 16.2 14.25
5 ½2:0; 2:0; 2:0 ½2:5; 4:0; 6:0 25.67 22.01 12.54 14.47 17.56 14.25
6 ½2:5; 2:0; 1:5 ½2:5; 4:0; 6:0 21.46 20.21 7.01 10.51 6.75 7.25
7 ½2:5; 2:0; 1:5 ½2:5; 3:5; 4:0; 4:5; 6:0 45.63 49.21 34.96 45.38 27.69 25.01

Table 2
Confusion (%) in training stage for the cooling scenarios.

Scenario P X PLAV PLSV SLAV SLSV NNA NNB

8 ½0:5; 0:5; 0:5 ½0:5; 1:0; 1:5 30.15 33.23 12.26 12.73 16.11 17.16
9 ½1:0; 1:0; 1:0 ½1:0; 2:0; 3:0 18.28 19.33 10.63 9.33 19.42 8.35
10 ½1:5; 1:5; 1:5 ½1:5; 3:0; 4:5 23.40 25.75 21.61 18.00 13.54 13.13
11 ½0:5; 0:5; 0:5 ½1:5; 2:0; 3:0 31.32 36.23 12.69 19.42 9.57 9.14
12 ½1:0; 1:0; 1:0 ½1:5; 2:0; 3:0 22.08 22.44 13.96 13.74 5.36 12.40
13 ½1:5; 1:0; 0:5 ½1:5; 2:0; 3:0 22.18 24.24 8.03 11.17 22.71 13.19
14 ½1:5; 1:0; 0:5 ½1:5; 2:0; 2:5; 3:0 46.53 46.84 39.19 44.61 26.84 23.38

100
C=6.0 kW 100
90

80 90

70 C=4.0 kW 80
60
QoS(%)

QoS(%)

50 70

40 C=2.0 kW 60
30
50
20

10 Sigmoid Function
Polynomial Function
40
0
-20 -15 -10 -5 0 5 10 15 20 30
Temperature(°C) PLAV PLSV SLAV SLSV NN_A NN_B

Fig. 7. Comparison of sigmoid and polynomial areas for scenario 2. Fig. 9. QoS test results for scenario 2 (heating).

100 24

90 23.5
80
23
70
22.5
60
QoS(%)

T(°C)

50 22

40 21.5

30
21
C=2.5 kW
20 C=3.5 kW
C=4.0 kW 20.5
10 C=4.5 kW
C=6.0 kW
20
0
-20 -15 -10 -5 0 5 10 15 20 25 PLAV PLSV SLAV SLSV NN_A NN_B
Temperature(°C)
Fig. 10. Average room temperature test results for scenario 2 (heating).
Fig. 8. Training data for scenario 7 (heating).
 J.A. Gomez, M.F. Anjos / Applied Energy 191 (2017) 492–501 499

100 29

90 28

27
80
QoS(%)

26
70

T(°C)
25
60

24
50

23
40

22
30
PLAV PLSV SLAV SLSV NN_A NN_B
PLAV PLSV SLAV SLSV NN_A NN_B
Fig. 11. QoS test results for scenario 7 (heating).
Fig. 14. Average room temperature test results for scenario 14 (cooling).

24 and 4:5 are not clearly defined. We also saw that different capaci-
ties can result in a similar QoS at the same temperature due to the
22
load shifting. Nevertheless, eliminating one of the classes can have
20 negative effects on the final output; in this case the NNs tend to
eliminate class 4:5 in order to minimize the confusion value.
18
Although C ¼ 4:5 and C ¼ 4:0 can accept two out of the three loads
16 if all of them arrive at the same time, the situation changes when
T(°C)

the loads arrive at different times. For example, C ¼ 4:5 will satisfy
14
any of the combinations of two loads arriving simultaneously:
12 ½2:0; 2:5; ½1:5; 2:5, and ½1:5; 2:0, whereas C ¼ 4:0 will not accept
½2:0; 2:5. It is therefore preferable not to eliminate a class because
10
of the dynamics in the system.
8

6
3.3. Results for fifty-load instance
4
PLAV PLSV SLAV SLSV NN_A NN_B
To demonstrate the scalability of the proposed methodology,
Fig. 12. Average room temperature test results for scenario 7 (heating). we present results for an instance with 50 space heaters. This
instance represents an apartment building with three different
types of heaters P ¼ ½1:5; 2:0; 2:5 with respectively 20; 15, and
15 loads of each type. We consider the scenario in which the build-
100
ing operator chooses X ¼ ½25:0; 45:0; 70:0; 90:5. Figs. 15–17 give a
summary of the results.
90

80 100
C=90.5 kW
90
QoS(%)

70
80
C=70.0 kW
60 70

50 60
QoS(%)

C=45.0 kW
50
40
40
C=25.0 kW
30 30
PLAV PLSV SLAV SLSV NN_A NN_B
20
Fig. 13. QoS test results for scenario 14 (cooling).
10

0
In the case of scenarios 7 and 14 we observe a special situation: -20 -15 -10 -5 0 5 10 15 20 25
although the training results for the NNs are better we have a
Temperature(°C)
worse QoS (Figs. 11 and 13) and temperature management (Figs. 12
and 14). We previously saw in Fig. 8 that the areas for classes 3:5; 4, Fig. 15. Classification areas.
500 J.A. Gomez, M.F. Anjos / Applied Energy 191 (2017) 492–501

100 The shaving effect can be achieved, controlling the peak consump-
tion, respecting the QoS, and ensuring a better utilization of the
90
power capacity available.
80 The proposed method computes capacity profiles for a specific
comfort zone with a defined set of loads. For different configura-
70 tions of the building and/or different boundary conditions, the user
can easily compute the new classification areas for different sce-
60
narios and intervals of comfort. The quality of the historical data
C(kW)

50 and coherence in the thermal system when defining the set of

loads are key to the applicability of this method.
40
Future work will explore the applicability of the proposed
30 methodology to more complex systems with different types of
buildings and loads and also take into account the user behavior.
20 Finally, the approach presented is computationally efficient, it
10 utilizes data that is normally available in the smart building con-
text, and it performs well for heating and cooling, offering better
0 performance than NNs in a real-world-based scenario.
-20 -15 -10 -5 0 5 10 15 20 25
Temperature(°C) Acknowledgments
Fig. 16. Capacity as function of external temperature.
We thank the editor and the two anonymous referees for their
detailed and helpful comments on earlier versions of this paper.
This research was supported by the Canada Research Chair on
100
23.5
Discrete Nonlinear Optimization in Engineering.
90
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