MODELING DEMAND-PRICE CURVE: A CLUSTERING APPROACH TO DERIVE
DYNAMIC ELASTICITY FOR DEMAND RESPONSE PROGRAMS
Ioannis P. Panapakidis, Department of Electrical Engineering, Technological Educational Institute of Thessaly, 41100 Larisa, Greece,
+302410684303, [email protected]
Athanasios S. Dagoumas, Energy & Environmental Policy lab., School of Economics, Business & International Studies, University of
Piraeus, 18532 Piraeus, Greece, +302104142651, [email protected]
Abstract
Demand Side Management (DSM) refers to a set of measures that aim at modifying the energy demand in periods
where the cost of generated electricity is high or the stability of the grid is jeopardized. The pillars of demand side
management are energy efficiency and demand response. Energy efficiency refers to a set of measures that refer to
the replacement of the equipment of electricity consumption or lowering the losses of the existing. The aim is to
provide the same energy services to the consumers under the less energy consumption. Demand Response (DR)
includes as set of actions that target on the modification of demand using efficient resources such as deferrable loads,
economic programs, incentives and others. Price-based DR programs aim to the modification of energy consumption
patterns through a set of tariff schemes such as time-of-use rates, real time pricing and others. Based on contracts
between the retailer and consumers, a special tariff is offered to the consumers that reflects more accurately the cost
of generated electricity in the day-ahead market. The consumers` response to the offered price is modeled via various
demand-price curves. A key factor that defines the level of adaption of the offered price is the elasticity that reflects
the flexibility of the demand. In the majority of DR programs, the elasticity is regarded a constant value within the
day. This approach does not accurately represent the potential of altering the demand pattern in hourly basis. The
scope of this study is to present a novel method to dynamic elasticity curves. Under this concept, the elasticity
receives different values and hence, a more robust design, implementation and assessment of DR programs can take
place.
Keywords: Clustering, demand function, demand modeling, demand response, elasticity
Introduction
Modern power system planning includes a variety of resources to meet the demand needs taking into account the
technical, economic and environmental constrains. A considerable body of research is focused on the smart grid`s
role in the contemporary power systems [1]-[4]. The main aim is the optimized collaboration between the electricity
generation schemes and the delivery of reliable and effective energy services to the end-consumers. The smart grids
coordinate the actions of the producers, system operators, consumers and others in order for the existing grid
infrastructure to operate optimally within various techno-economic and environmental constrains.
The ongoing adaptation of smart grid technologies results in several transformations in contemporary power systems
operation and planning. Smart grids promote the methods of altering the demand patterns of the various consumer
categories [5]. Special focus is placed in DR programs [6]. The concept of DR has risen due to the need of managing
more efficiently the demand patterns of the various consumer categories. The benefits of successfully implemented
DR measures have been a concept for discussion in numerous researches [7]-[8]. It is fact that many utilities, system
operators and retailers have an interest of providing the motives to the consumers to alter their patterns of electricity
usage for the purpose of lowering generation costs, network congestion an optimal distribution generation and
storage technologies [9]. Demand response can be seen as the potential that can transform every flexible consumer in
an active entity that can take part in the retail market. DR aims at significant demand savings by the combination of
load management techniques, distributed generation and storage. The exploitation of the benefits of DR is held via
numerous research studies in recent years. Also, various pilot programs have implemented in some European
countries [10].
DR programs include a variety of services and resources to accomplish the pre-defined goals of load modification
and management. Price-based DR includes dynamic pricing schemes are voluntary from a consumer perspective.
Real-Time Pricing (RTP) has the potential of transferring the actual cost of generated electricity to consumers. The
consumers modify their consumption according to the offered price. Also, dynamic pricing such as critical peak
pricing can lead to load reductions in peak periods in order to secure adequate generation capacity and lower
network congestion. Usually, price-based DR programs are implemented by a retailer or utility [11]. In [11], DR is
the basis of a profit maximization problem in day-ahead market. The consumers` response to the real-time prices is
modeled by different demand/price functions. These functions can be linear, logarithmic, exponential and others. The
crucial factor that determines the flexibility of the demand for modification (i.e. overall reduction, peak shifting,
peak shaving and others) is the elasticity parameter. The elasticity is defined as the level of demand alternation with
respect to the price signal send by the retailer or utility. It refers to the ratio of the load difference (i.e. initial load
minus the load after the DR) to the price difference (i.e. the initial price minus the price that caused the DR) and
determines the relationship between the price and the responsive demand as modeled by the demand function. In
[12], the authors check the sensitivity of the profits under scenarios of elasticity variation, discount on tariffs and
others. Dynamic pricing is also considered in [13], where the consumers` response is simulated by a linear demand
function. Other studies involving DR focus mostly at the procurement mechanism of the retailer [14]-[16]. The
retailer acts as the intermediate agent between the generation and demand sides. Apart from the selling prices that
determine the revenues, the cost of purchasing electricity from the wholesale market is another factor that influences
the profits. In [14], the authors formulate a mixed-integer stochastic programing problem that seeks the optimal
procurement mechanism (i.e. forward contracts and pool market) and selling price. Authors of [15] examine a riskconstrained stochastic programming framework to decide the price together with the type of forward contracts. The
optimal contract design in respect to prices and quantities, both at supply and end-user levels is addressed in [16].
All the previous studies consider the elasticity either constant within the day or consider different values per period,
namely off-peak and on-peak hours. Usually, it is given a fixed price that is not justified or based on preliminary
economic studies. As discussed in [17], empirical estimates of the real-time elasticity are hardly available. Long-term
elasticities refer to periods greater than a year while short-term elasticities cover periods up to one year. According
to various empirical studies reported in [17], short and long-term elasticities usually differ. Also, elasticities of offpeak and on-peak hours may present a low correlation. Finally, different energy sectors (i.e. residential, industrial,
etc.) correspond to various long-term elasticities [18]-[19].
The present paper presents a novel methodology for deriving hourly elasticities values that can be exploited in pricebased DR programs. We consider a simple theoretical case with a high voltage industrial consumer that purchases
electricity through a retailer in competitive retail market environment. The consumer is charged with a RTP scheme.
We take into account an existing industrial consumer connected to the high voltage grid in Greece. The period of
study is a complete year. According to the reports of Regulatory Authority of Energy S.A. of Greece, high voltage
industrial consumers are energy intensive [20]. Therefore, the potential of implementing DSM measures is high.
According to the respective legislation, all the HV consumers in Greece are monitored since they are eligible for
application of primary DSM actions, i.e. pre-negotiated load reductions during transmission system peaks. The view
that DSM potential is considerable in the industrial sector is also expressed in [21]-[22]. According to [21], the
interest of DSM in the industrial sector is high and the various benefits for the involved stakeholders are evident
using the appropriate business models. Authors of [22] argue that DSM is applicable in different industrial sectors
(i.e. steel, cement and others) targeting at different processes.
The proposed method is built on the utilization of clustering algorithms [23]. Clustering is a robust un-supervised
machine learning tool that can be efficiently used in a variety of applications and engineering problems [24].
Clustering is suitable in cases where no a priori knowledge of the data classes is available. It is a purely data-driven
process. Given a data set with limited or absent formal knowledge about the possible structure and the interrelationships among the data patterns, clustering seeks to identify a descriptive model. The initial data set can
represented with a reduced set of typical patterns or profiles. In the present paper, hourly System Marginal Price
(SMP) and load pair values serve as inputs of the clustering. Clustering is applied separately in hourly basis and the
profiles or representative patterns of the clusters are drawn. We select a specific profile and employ it for the
estimation of hourly elasticity. For the purpose of the detailed evaluation of the proposed method, three different
algorithms are compared, namely the K-means, Wards algorithm or Minimum Variance Criterion algorithm and the
Self-Organized Map (SOM). The algorithms differ in terms of efficiency, computational speed, input parameter
requirements and complexity. Also, two different demand functions are considered to examine the degree of
dependence of the hourly variation of elasticity and the used model (i.e. function) between the SMP and the load.
The method developed in the paper is characterized by simplicity and flexibility; different algorithms and demand
functions can be tested. Also, there is no restriction on the period under study; the method can applied to derive
daily, seasonal or annual dynamic elasticity curves. The dynamic elasticity curves drawn by the proposed method can
be utilized in empirical studies, retailer or/and consumer profit maximization problems, DR applications and others.
Method description
The available data sets consist of the load of the industrial consumer and the SMP of the Greek pool market. The
data expand to a full year. Fig.1- Fig.2 display the two time series, respectively. Let p0 (h) be the hourly price offered
to the consumer (€/kWh) in regulated market environment. In the present paper, p0 (h) corresponds to the SMP. Let
p(h) be the hourly price offered to the consumer (€/kWh) due to market competition. This price refers to a pricebased DR program such as RTP. In the present study, p(h) corresponds to RTP set by the retailer. For instance, for
simplicity reasons, the RTP can be 20% higher than the SMP. Also, let d0 (h) and d (h) be the initial and the final
load (kW) per hour, respectively. Load d (h) corresponds to the response due to the acceptance of price p(h). The
simplest representation between price and responsive demand is expressed via the linear demand function
d lin (h)  alin  blin p(h), where alin and blin are two coefficients of the linear function. Let BC (d lin (h)) and B0lin (h) be
the consumer`s obtained benefit (€) from consuming the hourly demand d lin (h) and the initial consumer`s benefit
(€) that corresponds to the initial load d0lin (h), respectively, when considering the linear demand function. The
consumer`s benefit from the use of electricity corresponding to linear demand function is given by:
 d lin (h)  d0lin (h) 
(1)
BC (d lin (h))  B0lin (h)  p0 (h)  d lin (h)  d 0lin (h)  1 

2 E lin (h)d0lin (h) 

where d0lin (h) and d lin (h) are the initial and the final load (kW) per hour that refer to the linear relationship between
load and price, respectively and E lin (h) is the hourly elasticity using the linear model. Moreover, we consider the
logarithmic demand function d lin (h)  alog  blog ln( p(h)), where alog and blog are two coefficients of the logarithmic
function. Let BC (d log (h)) and B0log (h) be the consumer`s obtained benefit (€) from consuming the hourly demand
d log (h) and the initial consumer`s benefit (€) that corresponds to the initial load d0log (h), respectively, when
considering the logarithmic demand function.
Fig. 1. Hourly load time series of the industrial consumer.
Fig. 2. Hourly SMP time series of the Greek wholesale energy market.
The consumer`s benefit from the use of electricity corresponding to logarithmic demand function is given by:
 d log( h )  dlog0 ( h ) 
(2)
B (d (h))  B (h)  p0 (h)d (h) E (h)  e E ( h ) d0 ( h )  1 




where d0log (h) and d log (h) are the initial and the final load (kW) per hour that refer to the logarithmic relationship
log
C
log
log
0
log
log
log
between load and price, respectively and E log (h) is the elasticity considering the logarithmic model. Using the linear
model, the consumer`s maximum benefit would be determined by
BC (d lin (h))
 0. By solving (1), we obtain the
d lin (h)
responsive load considering the linear model:

p(h)  p0 (h) 
d lin (h)  d0lin (h) 1  E lin (h)

p0 (h)


Accordingly, setting
(3)
BC (d log (h))
 0 and solving (2), we obtain the responsive load considering the logarithmic
d log (h)
model:

 p ( h)  
(4)
d log (h)  d0log (h) 1  E log (h) ln 
 

 p0 (h)  

The main definition of price elasticity implies differentiating the demand vs. price function. Thus, if a consumer’s
demand is modeled by a mathematical function, its price elasticity will be also defined based on the adopted model
referring to the following expression:
p0 (h)d (h)
(5)
d0 (h)p(h)
Differentiating a demand function leads to the hourly price elasticities according to the adopted demand model. By
substituting (5) in (3) and (4), the dynamic elasticities are obtained:
E ( h) 
E lin (h)  blin
p0 (h)
alin  blin p0 (h)
(6)
E log (h) 
blog
 blog ln( p0 (h))
(7)
alog
According to (6) and (7), when p0 (h) is known and by estimating the coefficients a and b, we obtain elasticity
values that change per hour [25]-[27].
For the purpose of deriving E (h) a clustering based approach is proposed. The proper representation of the data that
will feed the clustering algorithms is necessary. The clustering is applied on the hourly load and SMP pairs, i.e.
{d0 (h), p0 (h)}. Therefore, we refer to term pattern as a finite vector of load and SMP values. The dimension of the
vector
is
2
(i.e.
equals
the
number
of
elements).
The
set
of
the
pattern
is
denoted
as
X  {xm (h)  (d0m (h), p0m (h)), h  1,..., H , m  1,..., M }, where M indicates the number of the patterns. In the present
study we have H=24 and M=365. Note that the clustering is applied separately in each hour. The clustering process
is a mapping of M→ K, where K is the number of clusters and 1  K  M . Each cluster has a centroid which refers to
the mean of the patterns that belong to the cluster. The set of the clusters is denoted as
Ck  {ck (h), h  1,..., H , k  1,..., K}. The output of the clustering procedure is the formulation of the centroids.
After the data representation the next step is the selection of the clustering algorithm. A suitable algorithm should be
fast, simple and efficient. There are many algorithms that have been proposed in the literature applied on a diverse
set of applications. The algorithms can be divided in various categories, i.e. graphic-based, hierarchical, partitional
and others [23]. Each category approaches the clustering problem differently. In order to provide a reliable clustering
framework for the problem under study, a comparative analysis is held with different types of algorithms. It should
be noted that algorithms comparison is a common approach in pattern recognition problems within different
scientific fields. In the present study one algorithm per the most commonly used category is considered: K-means,
Ward`s algorithm or Minimum Variance Criterion and SOM.
K-means is the most common partitional clustering algorithm. The algorithm is based on a cost function
minimization process via a series of iterations. The flowchart of the K-means is presented in Fig. 3. The core of the
function of the algorithm is the minimization of an objective function, which is the Sum of Squared Errors (SSE).
The problem is formulated as follows:
K
Minimize:
M
SSE   umk d Eucl (ck , xm )
(8)
k 1 m 1
K
Subject to:
u
k 1
mk
 1,
1 m  M
(9)
umk  {0,1}, 1  m  M , 1  k  K
where the binary variable umk is an indicator whether the mth pattern belongs to cluster Ck  umk  1 or
not  umk  0  and the operator d Eucl refers to the Euclidean distance. The K-means algorithm consists of the
following steps:
Step1. Initialization. Start the algorithm with a random pick of K centroids from the subset Ck  X .
Step2. Clustering. At each iteration I , define each Ck as follows: for each m  1,..., M assign xm to Ck when
Euclidean distance d Eucl ( xm , ck ) is the smallest among the distances between xm and the rest of centroids.
Step3. Update. The new centroids of each cluster are calculated as the averages of the patterns of the same cluster.
Step4. Termination. The algorithm stops if there is no change in the partition at the I-th iteration; otherwise we
increment I to and I+1 repeat Steps 2 and 3 [28].
The Ward`s algorithm belongs to the category of the hierarchical agglomerative clustering algorithms. Hierarchical
clustering is not based on cost minimization and there are no iterations and transpositions of the patterns among the
clusters during each iteration. Hierarchical clustering starts with M singleton clusters, i.e. each pattern is considered
a sole cluster. The M×M proximity matrix is built and the minimum distance between the patterns is tracked. The
corresponding patterns are merged to form a new cluster. The matrix is updated and the distances are re-computed.
The process terminates when one cluster is left where all patterns are gathered into a single cluster. The user needs to
set the merging termination step prior to the formation of the single cluster. The merge of a pair of patterns depends
on the distance function between the current pattern and the newly formed one. Fig.4 shows the steps of hierarchical
agglomerative clustering. The agglomerative sequence is described in the following:
Step1. Starting from an initial population of M singleton clusters, the M×M proximity matrix is computed.
Step2. Let Cm , Cl be two different random singleton clusters of the data set. The minimal distance within the matrix
is derived:
d Dist (Ci , C j )  min d Dist (Cm , Cl )
1 m,l  M
(10)
where d Dist is the distance function and Ci , C j are the clusters which are merged to form a new cluster Cij . The
distance between Ci and C j is the minimum distance between all the random pairs Cm , Cl .
Step3. The proximity matrix is updated by calculating the distances between the cluster Cij and the others.
Step4. The steps 2 and 3 are repeated until one cluster is left.
The merge of a pair of clusters depends on the distance function between the cluster C j and the newly formatted
one, Cij . The general form of the distance function is given by:
d Dist (Cl ,(Ci , C j ))  ai d Dist (Cl , Ci )  a j d Dist (Cl , C j )   d Dist (Ci , C j )   d Dist (Cl , Ci )  d Dist (Cl , C j )
(11)
where ai , a j ,  and  are coefficients which define the form of the distance. For the Ward`s algorithm, we have
ai 
Mi  Ml
,
Mi  M j  Ml
aj 
M j  Ml
Mi  M j  Ml
,

M l
and
Mi  M j  Ml
  0, where M i , M j and M l represent the
populations of the clusters Ci , C j and Cl respectively [23].
The SOM is an unsupervised learning neural network that consists of a layer of neurons that are arranged in a
geometrical topology such as a line, plane or other. The SOM is trained through a number of epochs. One epoch
refers to the period needed for feeding all the patterns into the input layer. The layer can be one-dimensional or a bidimensional lattice. Every neuron is connected via weights with the input layer and receives a complete copy of the
input pattern. SOM is trained through competitive learning. The output unit with the highest activation towards the
input pattern wins the competition. The output units or neurons of the competitive layer or clusters` centroids are
updated each time a pattern is presented in the competitive learning. More specifically, a neuron affects positively
the neighboring neurons and negatively the most distant ones. When the layer is supplied with an input, a
competition between the neurons takes place; which neuron will respond more to the specific input. Through neurons
competition, a map is created on the topology of the network. The map resembles the grouping of the input patterns.
Specific regions on the map represent the groups. This means that the SOM projects the multi-dimensional input data
in one or two dimensional space. The flowchart of the SOM operation is presented in Fig.5. A weight updating takes
place for the winner neuron:
wk (t  1)  wk (t )   (t )hc( k ) (t )[ xm (t )  wk (t )]
(12)
where t is the respective epoch (i.e. the training step), wk (t ) and wk (t  1) are the weights of the winning neuron i at
epoch t and t  1 , respectively,  (t ) , hc( k ) (t ) and xm (t ) are the learning rate, the neighbourhood kernel around
neuron wk (t ) and the random input vector at time t , respectively.
For each hour h, the three algorithms are applied separately on the patterns xm (h) . The algorithms are run for
variable of clusters k and k profiles are formed. Therefore, the M=365 patterns per hour (h=1,…,24) are grouped in k
clusters. We kept the same k for each hour. Among the k profiles, we selected the one that corresponds to the most
populated cluster, i.e. the one that includes the most pattern of the initial population M. Hence, we have only 1
profile per hour. Next, by applying a regression fitting technique between the two elements of the profile (i.e. the
average {d0 (h), p0 (h)} of the most populated cluster the coefficient a and b are obtained. More specifically,
employing a linear function regression model the coefficients a lin and blin are obtained while with a logarithmic
function regression model the coefficients a log and blog are obtained. Then using the hourly SMP values (i.e. p0 (h) )
the elasticities E lin (h) and E log (h) are drawn using (6) and (7) [30].
Start
Selection of the
initial centroids
Centroid
change bellow
minimum?
Yes
No
Yes
All patterns
considered?
No
No
All centroids
considered?
Yes
Calculation of
the distances
Assign the patterns
into clusters
Calculation of the
new centroids
No
Iteration>
maximum?
Yes
End
Fig.3. Flow-chart of the K-means algorithm.
Results
The three clustering algorithms are applied separately on the set X . The scope of the comparison is to examine
whether the selection of the type of the algorithm influence the elasticity values and shapes. Prior to the operation of
the algorithms, the required input parameters should be selected. The selection can be done either by a parametric
analysis, i.e. to examine whether a parameter influences the clustering outcome or by incorporating external expert
knowledge. The parametric analysis refers to a set of trial-and-error experiments that check the sensitivity of the
clustering outcome by a crucial input parameter. In the present paper, we followed this approach. The complexity of
the algorithms as expressed by the input parameter requirements would be a considerable factor in big data sets. In
the problem under study, this is not case; the patterns dimension is relatively small. Also, the complexity can be
expressed by the required execution time. The most time demanding algorithm is the SOM, followed by the Kmeans. The required time refer to few seconds when a modern pc system is used.
Start
Singleton
clusters
Compute
distance matrix
Number of
clusters =1?
No
Merge two
closest clusters
Yes
Update distance
matrix
End
Fig.4. Flow-chart of the Ward`s algorithm.
Start
Type of map selection
Weight initialization
Enter pattern
into the network
Winning neuron
determination
Weights update
No
Termination
criterion met?
Yes
End
Fig.5. Flow-chart of the SOM algorithm.
Regarding the K-means, the parameters that need to be determined are the maximum number of iterations and the
minimum amount of improvement of the objective function between two successive iterations is met. The maximum
number of iterations is set to 500 and the minimum improvement to The Ward`s algorithm is less complex; it needs
only the merging stopping criterion between the consecutive merges. Actually, the merging stopping criterion is the
number of the clusters that need to be set by the user. The parameters of the SOM are: type of the map (i.e. one or
two dimensions), training epochs, initial weights selection, initial neighbourhood size and initial learning rate.
Moreover, we consider one dimension maps, i.e. {1, K}, where K is the number of clusters. The training epochs
equals to 500 and the initial weights are set to random values. Finally, the initial neighbourhood size is set equal to 2
and the initial learning rate is set equal to 0.10.
For each hour, we examined four different cases that corresponds to clusterings with k=5, k=10, k=15 and k=20.
Note that the optimal number of clusters is a parameter to properly be selected in many clustering applications since
it influences the quality and exploitability of the results. The optimal number can be drawn using a proper
mathematical criterion which refers to a clustering validity indicator. Apart from evaluating the performance of an
algorithm, the validity indicator can be used to denote the optimal number of clusters. The consideration of validity
indicator is beyond the scopes of the present work and will be examined in future research by the authors. Fig.6
presents two illustrative examples of clusterings with k=5. The data used refer to hour#11 and hour#21. The red dots
in the figures correspond to the profiles. It is observed that in both hours, K-means and SOM lead to almost similar
results.
Fig.6. Data and profiles of hour#11 obtained by: a) K-means, b) Ward`s algorithm, c) SOM and data and profiles of
hour#21 obtained by: d) K-means, e) Ward`s algorithm, f) SOM.
More specifically, hour#11 profiles produced by the K-means have the following coordinates: {1564, 0.0682},
{5370, 0.0465}, {1825, 0.0523}, {3804, 0.0435} and {315, 0.0563}. The profiles of the SOM are: {1825, 0.0523},
{5370, 0.0465}, {1825, 0.0523}, {3804, 0.0435} and {315, 0.0563}. The two algorithms provide the four similar
profiles. In the problem under study, partitional clustering represented by the K-means results in similar operation
with vector quantization clustering represented by the SOM. This is mainly due that these algorithms are built on
objective function minimization iterative process. However, this is not the case with Ward`s algorithm. Hierarchical
clustering tends to create considerably different clusterings [23]. The coordinates of the respective profiles are:
{5772, 0.0507}, {5093, 0.044}, {3718, 0.0437}, {1735, 0.0583} and {421, 0.057}. According to Fig.6c), the
profiles are most distinct from each other and they are more widespread in the coordinate space compares to the
profiles of the K-means and SOM.
Hour#11 presents some outliers, i.e. data points with extreme values. These refer to high value of load and SMP. No
profile is placed near to them. This means that they are included in other clusters and do not represent a cluster by
their own. Similar conclusions are drawn from the clusterings of hour#21 regarding the algorithms performance.
Here, the outliers are more frequent.
Fig.7 depicts the hourly elasticity values considering the K-means algorithm and the linear demand function. Results
for k=5, k=10, k=15 and k=20 are presented. Recall that elasticity is the parameter that determines how flexible the
demand is during the examined period. If the consumer is elastic, the modification of the demand is more evident and
follows the variation of the price signals. According to Fig.7, it can be noticed that the consumer presents various
degrees of demand flexibility through the year as modelled by the volatility of the dynamic elasticity curve. Values
close to 0 correspond to inelastic behaviour. However, due to Greek pool market operation, in several hours within
the year the SMP is 0. This leads to E lin (h)  0. For k=5, there are 125 instances with E lin (h)  0 that correspond to
p0 (h)  0, referring to the 1.42% of the total number of hours. A careful examination is needed to conclude whether
zero elasticity represents the inertia of the demand or it is due to zero SMP. While most consumers tend to change
the demand even in periods with high prices, zero elasticity comes directly from instances with p0 (h)  0. The
maximum absolute annual value is E lin (h)  0.2073. For clusterings with k=10, k=15 and k=20 the maximum
absolute values are 0.1984, 0.2372 and 0.2108, respectively. As for the average annual values, we have 0.0878,
0.0818, 0.0993 and 0.892 for k=5, k=10, k=15 and k=20, respectively. The different number of clusters does not
influence in a large degree the annual maximum and average values. Additionally, the elasticity curves` shapes do
not present large variations. Fig. 8 shows the maximum daily elasticity for the various numbers of clusters. It is
shown that the elasticity do not present a periodicity or a seasonal trend. Considering the curve for k=15, it can be
noticed that the maximum elasticity is, in general terms, increasing from the beginning of the year until the 106th
(i.e. 16/04/2010) and afterwards, for the next 40 days it presents a stable behaviour with the higher peaks. Also,
periods of high peaks are observed in the periods 17/08/2010-03/09/2010 (i.e. days 229-256), 27/09/201018/10/2010 (i.e. days 270-291) and 17/12/2010-19/12/2010 (i.e. days 351-353). The period 26/06/2010-07/08/2010
refers to a stable inelastic behaviour. During the summer period, the consumer is more inelastic due to the presence
of necessary universal service loads such as air-conditioners.
Fig.8-Fig.12 present the dynamic elasticities curves based on different clusterings. More specifically, Fig.8 shows the
dynamic elasticity using the K-means and logarithmic demand function. Here, instances with E log (h)  0 correspond
also to SMP values p0 (h)  0. The annual maximum elasticities for k=5, k=10, k=15 and k=20 are 0.3479, 0.3519,
0.3492 and 0.3449 in absolute values, respectively. The logarithmic model leads to higher elasticies. Also, another
difference with the linear model is that the highest elasticity is met at k=10 contrary to the linear model (k=15).
Again, the different number of clusters does not considerably influence the shape of the curve. However, large
differences are observed between the curves of the two models. The linear model results in curves with high
volatilities and lower peaks. Neglecting zero and other small values, the curve of the logarithmic model leads to a
stable low value of -0.25. This value is almost constant during the whole period. Fig.9 presents the results of the
SOM algorithm and the linear model. The shapes of dynamic elasticities present high correlation with the
corresponding of the K-means. The annual maximum elasticities for k=5, k=10, k=15 and k=20 are 0.2229, 0.2518,
0.2021 and 0.2059, in absolute values, respectively. The maximum elasticity is met at k=10. SOM leads to higher
values compared with the K-means. Also, Ward`s algorithm (Fig. 11) leads to similar shapes. The annual maximum
values are 0.2248, 0.2213, 0.2209 and 0.2039, in absolute values, respectively. These values are higher compared to
the ones of the K-means and are close to the SOM. In addition, similarities are observed between the shapes obtained
by the logarithmic model. Therefore, we can conclude that the shapes of the curves are mostly influenced by the
selection of the type of the demand function and in a lesser degree by the number of clusters. Fig.13 present the
average daily elasticity curves for k=5. According to Fig.13a) the more elastic behaviour is met at the periods 11:0014:00 and 20:00-22:00. These correspond also to the morning and evening peak periods of the Greek interconnected
system. The consumer is more elastic in periods of increased demand and therefore, DR measure targeting on peak
shaving will be of greater interest. Similar conclusions are drawn using the logarithmic function. However, in this
case the three algorithms lead to almost identical elasticity shapes.
Fig.7. Dynamic elasticity curves obtained by the K-means algorithm and the linear demand function for: a) k=5, b)
k=10, c) k=15 and d) k=20.
Fig.8. Maximum dynamic elasticity curves obtained by the K-means algorithm and the linear demand function for: a)
k=5, b) k=10, c) k=15 and d) k=20.
Fig.8. Dynamic elasticity curves obtained by the K-means algorithm and the logarithmic demand function for: a)
k=5, b) k=10, c) k=15 and d) k=20.
Fig.9. Dynamic elasticity curves obtained by the SOM algorithm and the linear demand function for: a) k=5, b)
k=10, c) k=15 and d) k=20.
Fig.10. Dynamic elasticity curves obtained by the SOM algorithm and the logarithmic demand function for: a) k=5,
b) k=10, c) k=15 and d) k=20.
Fig.11. Dynamic elasticity curves obtained by the Ward`s algorithm and the linear demand function for: a) k=5, b)
k=10, c) k=15 and d) k=20.
Fig.12. Dynamic elasticity curves obtained by the Ward`s algorithm and the logarithmic demand function for: a) k=5,
b) k=10, c) k=15 and d) k=20.
Fig.13. Mean daily dynamic elasticity curves obtained by the: a) linear and b) logarithmic demand function.
Conclusions
The elasticity parameters hold a crucial role in the success of economic models that simulate the response of the
consumer to price signal. In the majority of the related literature of retailer profit maximization problems that involve
the demand response of the consumers, the value of elasticity is considered constant through the day period.
However, this approach does not reflect always the actual behaviour of the consumers. The demand is more elastic in
various periods through the day and through the various seasons. In this paper, a novel method is proposed to extract
dynamic elasticity curves. The clustering tool is utilized to extract the profiles of demand-price patterns through the
day. The profiles are used to determine the hourly elasticity of a complete year. The proposed method leads to the
extraction of a different elasticity value per hour, resulting in a more accurate modeling of the consumer` s
responsiveness to the price signals. The proposed method is applied to a high voltage industrial consumer located in
Greece. The main conclusions drawn from the algorithms application can be summarized in the following:


No strong correlation is observed between the shape of the dynamic elasticity and number of clusters. This is
mainly due to the consideration of low dimension patterns. In higher dimension problems, the number of clusters
may be an influential factor and thus, a proper cluster validation mathematical criterion should be used.
The selection of the demand function defines the level and the volatility of the dynamic elasticity. Linear
function leads to lower values and volatile shapes while more smooth shapes are observed using the logarithmic
demand function.


The type of the algorithm does not largely influence the results. If complexity is significant factor, the Ward`s
algorithm is proposed.
The consumer is more elastic in periods with overall increased demand, i.e. interconnected system peaks.
Therefore, the potential of DR to shave or shift the consumer`s peak is evident.
The present study serves as an initial study in the examination of methods to derive elasticity values that are more
representative of the actual responsive behaviour of the industrial consumer. The method is characterised by
simplicity and flexibility and therefore it can be integrated in profit maximization problems or other economic
analyses.
References
[1] M. Wissne, “The smart grid - A saucerful of secrets?”, Applied Energy, vol. 88, iss. 7, pp. 2509-2518, July 2011
[2] H. Lund, A.N. Andersen, P.A. Østergaard, B.V. Mathiesen and D. Connolly, “From electricity smart grids to
smart energy systems – A market operation based approach and understanding”, Energy, vol. 42, iss1. 1, pp. 96102, June 2012
[3] S. Blumsack and A. Fernandez, “Ready or not, here comes the smart grid!”, Energy, vol. 27, iss. 1, pp. 61-68,
January 2012
[4] M. Moretti, S. Njakou Djomo, H. Azadi, K. May, K. De Vos, S. Van Passel and N. Witters, “A systematic
review of environmental and economic impacts of smart grids”, Renewable and Sustainable Energy Reviews,
vol. 68, Part 2, pp. 888-898, February 2017
[5] International Energy Agency. Technology Roadmap Smart Grids. Paris; IEA: 2011.
[6] M.H. Albadi and E.F. El-Saadany, “A summary of demand response in electricity markets”, Electric Power
Systems Research, vol. 78, no.11, pp. 1989-1996, November 2008
[7] M. Qadrdan, M. Cheng, J. Wu and N. Jenkins, “Benefits of demand-side response in combined gas and
electricity networks”, Applied Energy, vol. 192, pp. 360-369, 15 April 2017
[8] U.S. Department of Energy. Benefits of demand response in electricity markets and recommendations for
achieving them. U.S. DOE: Washington, U.S.A., 2006.
[9] P. Siano, “Demand response and smart grids-A survey”, Renewable and Sustainable Energy Reviews, vol. 30,
pp. 461-478, February 2014
[10] P. Bertoldi, P. Zancanella and B. Boza-Kiss, “Demand Response status in EU Member States”, JRC Science for
Policy Report, European Comission, 2016
[11] J.M. Yusta, H.M. Khodr and A.J. Urdaneta, “Optimal pricing of default customers in electrical distribution
systems: Effect behavior performance of demand response models”, Electric Power Systems Research, vol. 77,
iss. 5-6, pp. 548-558, April 2007
[12] J.M. Yusta, I.J.R. Rosado, J.A.D. Navarro and J.M.P. Vidal, “Optimal electricity price calculation model for
retailers in a deregulated market”, International Journal of Electrical Power and Energy Systems, vol. 27, iss. 56, pp. 437-447, June-July 2005
[13] N. Mahmoudi-Kohan, M. Parsa Moghaddam, M.K. Sheikh-El-Eslami and E. Shayesteh, “A three-stage strategy
for optimal price offering by a retailer based on clustering techniques”, International Journal of Electrical Power
and Energy Systems, vol. 32, iss. 10, pp. 1135-1142, December 2010
[14] A. R. Hatami, H. Seifi and M.K. Sheikh-El-Eslami, “Optimal selling price and energy procurement strategies for
a retailer in an electricity market”, Electric Power System Research, vol. 79, iss.1, pp. 246-254, January 2009
[15] M. Carrión, J.M. Arroyo and A.J. Conejo, "A bilevel stochastic programming approach for retailer futures
market trading", IEEE Transactions on Power System, vol. 24, no.3, pp. 1446-1456, August 2009
[16] S. A. Gabriel, A.J. Conejo, M.A. Plazas and S. Balakrishnan, "Optimal price and quantity determination for
retail electric power contracts", IEEE Transactions on Power Systems, vol. 21, iss.1, pp. 180-187, February
2006
[17] M.G. Lijesen, “The real-time price elasticity of electricity”, Energy Economics, vol. 29, iss. 2, pp. 249-258,
March 2007
[18] N. Wang and G. Mogi, “Industrial and residential electricity demand dynamics in Japan: How did price and
income elasticities evolve from 1989 to 2014?”, Energy Policy, vol. 106, pp. 233-243, July 2017
[19] R.Bernstein and R. Madlener, “Short- and long-run electricity demand elasticities at the subsectoral level: A
cointegration analysis for German manufacturing industries”, Energy Economics, vol. 48, pp. 178-187, March
2015
[20] Regulatory Authority of Energy (RAE). 2014 National Report to the European Commission. December 2014,
Athens, Greece.
[21] D. Khripko, S.N. Morioka, S. Evans, J. Hesselbach and M.M. de Carvalho, “Demand side management within
industry: A case study for sustainable business models”, Procedia Manufacturing, vol. 8, pp. 270-277, 2017
[22] C.-F. Lindberg, K. Zahedian, M. Solgi and R. Lindkvist, “Potential and limitations for industrial demand side
management”, Energy Procedia, vol. 61, pp. 415-418, 2014
[23] R. Xu and D. Wunsch, “Survey of clustering algorithms”, IEEE Transactions on Neural Networks vol. 16, no. 3,
pp. 645-678, May 2005
[24] P. D'Urso, “Informational Paradigm, management of uncertainty and theoretical formalisms in the clustering
framework: A review”, Information Sciences, vol. 400-401, pp. 30-62, August 2017
[25] S. Yousefi, M. Parsa Moghaddam and V.J. Majd, “Optimal real time pricing in an agent-based retail market
using a comprehensive demand response model”, Energy, vol. 36, iss. 9, pp. 5716-5727, September 2011
[26] M. Parsa Moghaddam, A. Abdollahi and M. Rashidinejad, “Flexible demand response programs modeling in
competitive electricity markets”, Applied Energy, vol. 88, iss. 9, pp. 3257-3269, September 2011
[27] H.A. Aalami, M. Parsa Moghaddam and G.R. Yousefi, “Modeling and prioritizing demand response programs
in power markets”, vol. 80, iss. 4, pp. 426-435, April 2010
[28] N. Nezamoddini and Y. Wang, “Real-time electricity pricing for industrial customers: Survey and case studies in
the United States”, Applied Energy, vol. 195, pp. 1023-1037, 1 June 2017
[29] D. Steinley D, “K-means clustering: A half-century synthesis”, British Journal of Mathematical and Statistical
Psychology, vol. 59, pp. 1-24, May 2006
[30] T. Kohonen. Self-organisation and associative memory. 3rd ed., Springer-Verlag: Berlin, Germany, 1989