IEEE TRANSACTIONS ON SMART GRID FINAL VERSION REVISED JULY 2016 1
Residential Demand Management using
Individualised Demand Aware Price Policies
Barry Hayes, Member, IEEE, Igor Melatti, Toni Mancini,
Milan Prodanovic, Member, IEEE, and Enrico Tronci.
Abstract—This paper presents a novel approach to Demand
Side Management (DSM), using an “individualised” price poli cy,
where each end user receives a separate electricity pricing
scheme designed to incentivise demand management in order to
optimally manage flexible demands. These pricing schemes have
the objective of reducing the peaks in overall system demand
in such a way that the average electricity price each individual
user receives is non-discriminatory. It is shown in the paper
that th is approach has a number of advantages and benefits
compared to traditional DSM approaches. The “demand aware
price policy” approach outlined in this paper exploits the knowl-
edge, or demand-awareness, obtained from advanced metering
infrastructure. The p resented analysis includes a detailed case
study of an existing European distribution network where DSM
trial data was available from the residential end-users.
NOMENCLATURE
Indices and sets
S set of substation indices
s indices of substations (elements of S)
T set of time-slots indices
t indices of time-slots (elements of T )
U set of residential user indices
u indice s of residential users (elements of U)
T tariff system
P set of individualised price policies
Pa rameters and Constants
d
t
u
forecasted power demand of user u in time-slot t
˜
d
t
u
historical power demand of user u in time-slot t
ˆ
d
t
u
power demand of user u in time-slot t, as a response
to individualised price policies
P
t
S
maximum power for substation s in time-slot t
Q
u
capacity of storage of user u
R
u
power rate of storage of user u
C
u
maximum power for user u, from energy con tract
α multiplier for C
u
, in order to guarantee minimum
low-tariff energy
b
l
residential users low price for buying energy
b
h
residential users high price for buying energy
β residential users price for selling energy from local
renewable sources
The authors are with the IMDEA Energy Institute, Avda. Ram´on de
la Sagra, M´ostoles Technology Park, Madrid 28935, Spain, and Sapienza
University of Rome, via Salaria 113, 00198 Rome, Italy. Corresponding author
e-mail: barry.hayes@imdea.org. The research leading to these results has
received funding from the European Union Seventh Framework Programme
(FP7/2007-2013) under grant agreement number 317761 (SmartHG) and
the Spanish Ministry of Economy and Competitiveness project RESmart
(ENE2013-48690-C2-2-R).
Variables
b
t
u
state of charge of storage of user u in time-slot t
a
t
u
charge/discharge action on stora ge of user u in time-
slot t
∆
t
exceeding power at substation level
P
t
u
maximum power for user u in time-slot t
ζ
u
percentage of load shifting required f or user u
ξ index of price policy non-discrimination
I. INTRODUCTION
T
His paper presents a set of Demand Side Management
(DSM) software services which were introduced in the
SmartHG projec t [ 1]. These services are designed to manage
residential end-user energy demand with two objec tives: to
minimize energy costs for each individual user, and to a ssist
the Distribution System Operator (DSO) in managing network
constraints and optimising the operation of the distribution
system. This is a chieve d by exploiting d e m and aware ness
as obtained from smart m etering and other Advanced Me-
tering Infrastructure (AMI). One of the unique aspects of
the approa ch introduced in the SmartHG pr oject is that the
price policies are “individualised”, e. g. each individual user
receives a separate electricity pricing scheme designed to
incetivise demand management in order to optimally manage
flexible demands. These pricing schemes are designed with the
primary objective of reducing the peaks in overall distribution
system dem a nd, which has significant benefits for the network
and for the DSO. This is achieved in such a way that the
average e le ctricity price each individual user receives is non-
discriminatory, and the pr ic ing policy is designed to shape the
demand without reducing the overall demand volume, which
is undesirable from the DSO point of view.
The paper is structured as follows: Section II discusses
previous work, Section III contains the problem formulation,
and Section IV outlines the methodology. Section V describes
the case stu dy a nd results, and Section V I concludes.
II. PREVIOUS WORK
In light of the increasing penetrations of variable energy
resources, and the decreasing contribution from traditional
generation sources, i.e. large, controllable thermal generation
plant, a number of recent studies have identified a need for
new sourc e s of flexibility in electr icity networks, e.g., [2] ,
[3]. For several decades, network operators have used various
forms of DSM to improve the bala ncing of system supply and
demand and to red uce load peaks. Many of th e p ractical DSM
IEEE TRANSACTIONS ON SMART GRID FINAL VERSION REVISED JULY 2016 2
programmes implemented worldwide to date have focused on
large industrial consumers since these have demand of suffi-
cient volume to produce significant effects at the system level
(e.g., [4], [5]). However, with the introd uction of smart meter-
ing and time- varying electricity rates f or individual customers,
new oppo rtunities are being created for small electricity users
to p articipate in demand side services. Recently, much resear c h
has focused on DSM in the residential sector [6]–[13].
Many existing DSM programmes use direct load contr ol,
where the network op e rator is able to directly actuate large
industrial loa ds according to the needs of the network [4],
[5], and the exact terms of the DSM contract are agreed
beforehand. While direct load control may be suitable for cer-
tain industrial users, it has technical and practical difficulties
in the context of residential users, where direct actuation of
loads is typically considered an invasion of user privacy and
comfort. In a ddition, direct load control may re quire large
investments in order to provide additional commun ic ation
and control technology for each user. Most residential DSM
schemes instead rely on the user response to a electricity pric e
signal to produce the required outcome, e.g., [6], [7], [9]–[14].
Recent studies carrie d out in Ireland [15] and in Den-
mark [16], have tested the response of resid e ntial users to
various Time of Usage (ToU) electricity pricing schemes, in
order to quantify their potential to offer DSM serv ic e s. It
was shown in these trials that a significant amount of the
residential d emand (up to 19% in [16]) could be shifted away
from the peak hour if appropriate economic incentives are
applied. In order to achieve the volumes of demand required to
participate in th e electricity market, and to m ake a significant
contribution to system-level energy balancing, a mea ns of
combining and coordinating DSM actions from many highly-
distributed users is required. Sever al approa ches for this have
been proposed, such as the “aggr egator” [17] and “virtual
power plant” concepts [18], [19].
One of the d rawbacks of traditional appro aches to DSM,
where all users are subject to the same price (globa l price
policy), is that the peak-shifting sch emes may result in un-
desirable “rebou nd” effects, e.g. simply shifting the demand
peaks from the peak hou r to the off-peak hours, and creating
new dema nd peaks, or “rebound peaks” [11]–[13]. The authors
in [11] and [12] discuss autom a te d DSM algorithms designed
to schedule flexible resid ential user loads. It is demonstrated
that th is can produce rebound peaks, and it is con c luded in [11]
that DSM algorithms need to be studied for large numbers of
devices and users, and that DSM schemes need to be d esigned
with the objective of flattening the overall electric ity usage in
order to avoid these issues.
Exposing electricity end users to wholesale market prices
(real-time pricing) has the drawback that demand may be
shifted to hours with low electricity price, which can “ lead
to a higher peak electricity price and peak-to-average ratio
during the low price time”, according to [13]. There is a
significant challeng e in ensuring that such real-time prices do
not cause physical or market instabilities [ 20], and it has been
shown that multiple, uncoordinated responses to frequently
changin g prices can cause increased volatility, and potentially
grid instability [21]–[23].
III. PROBLEM FORMULATION
A. Individualised User Price Policies
A real-world example of the rebound effect is shown below,
using recorded data from a residential DSM study ca rried out
by a Danish DSO [16] . In th is study, re sid ential users were
given a time-varying price policy with three distinct pricing
periods, d esigned to test the flexibility of residential demand
to economic peak-shifting incentives. T he values are provided
below in Danish Krone (DKK):
• Day: 1.50 DKK / kWh (06:00 -17:00)
• Peak: 8.00 DKK / kWh (1 7:00-20:00)
• Night: 0.00 DKK / kWh (20:00-06:00)
This electricity pricing scheme provid e d strong inc e ntives
for household users and assessed the potential for residential
users to shift their demand away from the peak cooking hours
(17:00-20:00) to the night period (20:00-06:00 ) where the
price is ze ro
1
.
The above price policy was applied to a “Test group” of
350 households, with a “Reference group” of 349 households
2
receiving a fixed price of 2.25 DKK / kWh at all hours
during the day (as per the standard flat tariff residential pricing
scheme used in Den mark). The study was carried out over a
full year from 1 October 2013 to 30 September 2014. The
results showed that, with the above pr ic e incentives, a signif-
icant amount of residential demand (up to 19%) was shifted
away from the peak h ours, compared to the reference group.
The p rice incentives had little effect on the overall demand
consumption, i.e. the total volume of consumption remained
almost constant, only the times at which consum ption occurred
were influenced by the pricing scheme.
2 4 6 8 10 12 14 16 18 20 22 24
0
5
10
15
20
25
30
35
Time of day [h]
Average monthly
consumption [kWh]
Reference group
Test group
Peak
Night
Night
Day
Fig. 1. Sample of the results from the Danish study, showing peak shifting
in the Test group compared to the Reference group during January 2014 [16].
Fig. 1 shows a sample of the results from the Danish
study [16] fo r the month January 2014. All months of the
year showed a similar pattern, but the amount of demand shift
from the peak hour was largest during the winter months.
The results in Fig. 1 are a good example of the “rebound
peak” effects whic h can result from g lobal ToU price policies.
1
It could be argued that this is a rather extreme pricing policy and that the
zero night time price is more likely to cause rebound peaks than a pricing
scheme with, for example, a non-zero night-time price. However, the results
of this study are insightful in that they demonstrate that significant residential
demand-shifting is possible with appropriate financial incentives, and these
results provide real data on user responses to price signals for the subsequent
analysis in the paper.
2
The SEAS-NVE study was originally designed with 350 test and 350
reference householders, but data was unavailable for one of the reference
group households; hence this group only has 349 households.
IEEE TRANSACTIONS ON SMART GRID FINAL VERSION REVISED JULY 2016 3
There is a significant increase in d emand at the beginn ing
of the off-peak period for the “Test group”, a nd this pricing
scheme has cre a te d a new demand pea k at 21:00, which is even
larger than the demand peak in the “Referenc e group”, Fig. 1.
These effects are undesirable, since typ ic al DSM objectives
are to smooth the load pr ofile, incr e ase the load factor, and
reduce demand peaks.
An alternative solution to the “global” price policy, where
all users (o r at least all users in the same sector), receive
the same price incentives is prop osed in this paper. Instead ,
an “individualised” price policy is provided to each user,
using the Demand-Aware Price Policy (DAPP) computational
service outlined in Section IV. This is designed to maximise
the benefits of demand response actio ns f or both the user
and the DSO. The effects on network operation and the
potential benefits to the DSO in using such an individualised
price policy are evaluated using the scenarios pr esented in
Section V.
Even with strong ToU price incentives for demand-shif ting,
such as those used in the SEAS-NVE study [16]), th e amou nt
of demand flexibility fr om residential appliances is limited.
Energy Storage Systems (ESS) offer much greater possibil-
ities for demand flexibility. It is widely expected that the
cost of such techn ologies will continue to decrease, mak ing
storage accessible to a wide rang e of users, includ ing domestic
consumers. Moreover, Plu g-in Electr ic Vehicles ( PEVs) can
significantly alter the dem and profiles, and create critical
congestions in the distribution network if their charging is not
managed app ropriately [24], [25]. Acco rdingly, this paper also
analyses scenarios where the residen tial homes are equipped
with PEVs and ESS, in order to examine potential future
scenarios with greater user flexibility in response to ToU
pricing.
IV. METHODOLOGY
A. Overview
The proposed methodology is based on two integrated soft-
ware services, which are described schem a tically in Fig. 2. The
Electricity Distribution Network Virtual Tomography (EVT)
service is aimed at assisting the DSO in the opera tion and
management of the distribution networks. The EVT service
uses available measur e ments from Superviso ry Control And
Data Acquisition (SCADA) a nd smart metering/AMI systems
to estimate the network state in real-time using a Weighted
Least Squares (WLS) state estimator, and also to carry out
network analysis ahead of time, as described in [26], [27]. This
detects possible violations of network constraints, and raises
warnings and alarms to the DSO accordingly. T he results of
the state estimation and network analysis carried out in the
EVT ca n be used to make oper a tional constraints, limiting
the demand drawn at some o r all substations s w ithin the
distribution network at times of pea k deman d. This could be
motivated b y e conomic reasons (e.g. in order to reduce the
cost for the D SO of buying energy from the market at times
of peak electricity price), by technical rea sons (in order to
reduce overloading of network components during tim e s of
peak demand, or dur ing period s when the system is weakened
due to line/transformer maintenance or other outages).
The second service, DAPP, is designed to redistribute the
power demand (load shifting) so that th e constraints on the
substations s are fulfilled. T his is done by computing, for each
residential home u connected to s, an individualised suggested
power profile P
u
(i.e. different users may get different power
profiles), so that if all users follow their power profile then the
operational constraints o n s, as suggested by E V T, are m et.
Each user u is motivated to follow the suggested power profile
P
u
by an individu alised price policy based on P
u
. Proposing
individualised pr ic e policies avoids the problem of rebound
peaks, i.e. where the demand peaks are sim ply shifted through
the day, which may violate operational constraints.
Fig. 2. The proposed services architecture.
B. The DAPP S ervice
In this section, the DAPP service is described. In orde r to
understand h ow DAPP works, we first define how we model
residential users flexibility (Section IV-B1). Then, w e ou tline
the DAPP service input-output behaviour (Section IV-B2).
Third, we formally define one of the main requir ements for the
DAPP service, i.e., the fact that resulting individualised price
policies must be non-discrimina tory (Section IV-B3). Finally,
we describe the algorithm underlying the DAPP service (Sec-
tion IV-B4) and we prove th at it outputs non-discrimina tory
price policies (Section IV-B5).
The notation used in the DAPP service is as follows: a time-
slots set T is a fin ite set of contiguous time-slots, all having the
same duration τ (in minutes). Without loss of generality, we
will assume T to contain time-slot indic e s. A power profile
is a function P : T → R, where we write P
t
for P (t). A
power pr ofile P
1
follows a power pro file P
2
if and only if
P
1
(t) ≤ P
t
2
for all t ∈ T . The a rea of a power pr ofile P on T
is τ
P
t∈T
P
t
, i.e., the overall energy yielded by P . Fina lly, a
Linear Programming (LP) pro blem is a minimisation problem
over a set of linear inequalities (constraints) on real variables.
1) Residential User Flexibility Model: In our approach,
each residential user u is provided, with a given period ic ity
(every day in our experiments), with an individualised price
policy to be followed. Such price policy is defined on th e
basis of an individualised power profile P
u
. The resulting tariff
for u, which we call DAPP tariff, is based on two prices for
energy, the high price and the low price: if user u needs power
˜
d
t
u
in time-slo t t, then u will pay the low price if
˜
d
t
u
≤ P
t
u
, and
IEEE TRANSACTIONS ON SMART GRID FINAL VERSION REVISED JULY 2016 4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
12 13 14 15 16 17 18 19 20 21
kW
Hours in day 2012-09-06
Historical demand
Shifted demand
Max demand for low tariff area
Fig. 3. Historical demand compared to power profile output by DAPP for a
single home on a given day in our evaluation scenario.
the high tariff otherwise. As a con sequence, we call low tariff
area the area of P
u
on T . Note that the DAPP tariff is: (i)
Inclining Block Rate (IBR) (two prices are used d epending on
user power demand ); (ii) ToU (P
u
varies with time); and (iii)
individualised (P
u
varies with the user too). As an example,
Fig. 3 shows the individualised power profile P
u
(red cur ve)
together with the actual power demand d
u
(green curve) o n
the 6th of September 2012 for a selected user u connected to
a selected substation s in the reference scenario we will use
for our experimental evaluation (see Section V). I n the time-
slot t
1
from 16:00 to 17:00, the user is outside the low tariff
area (i.e., d
t
1
u
> P
t
1
u
), thus the high ta riff is applied, whilst in
the preceding time-slot t
2
the low tariff is applied. In order to
stay inside the low tariff area also in t
1
, the user shou ld be
flexible and, as an example, move approximately 0.3 kW of
power demand (i.e., d
t
1
u
− P
t
1
u
) from t
1
to t
2
(load shifting).
In this section, we want to provide a mathematical mo del
for user flexibility, based on the one in [14], in order to show
the effectiveness of the DAPP-based methodology in scenarios
in which the flexibility of each user may vary. To this aim,
we proceed as f ollows. We model flexibility of a residential
user u by means of a load shifting capability. Such capability
may be either “ physical”, i.e., user u has a battery installed at
home, or “vir tual”, i.e., user u has to shift the loads of other
appliances, in order to stay inside the low tariff a rea.
Given this, we model the load shifting capability, both in
the physical and in the virtual case, as an E SS: the power rate
of the ESS defines a bound on load shifting in each time-slot,
whilst the capacity of the ESS gives a bound on the summatio n
of consecutive load shifts. Namely, in our mathematical model,
the flexibility of u is a pair (Q
u
, R
u
), where Q
u
is the ESS
capacity (i.e., the maximum ene rgy which may be stored, in
kWh) and R
u
is the ESS power rate (i.e ., the maximum power
in kW which may be used from or saved into the ESS in a
given time-slot). In the example of Fig. 3, in order to perf orm
the above described load shifting, it is sufficient to have Q
u
=
0.3 kWh and R
u
= 0. 3 kW. Since the user has to move 0.3 kW
in on e hour, from time-slot t
1
to time-slot t
2
, the ESS must be
charged by 0.3 kW in time-slot t
1
and discharged by 0.3 kW in
time-slot t
2
. Both actions require at least Q
u
= 0.3 kWh and
R
u
= 0.3 kW. Of course, the ESS modelling the load shifting
capability should have been c harged during tim e-slot t
2
(from
15:00 to 16:00, where the user needs less energy than the one
allowed in low tariff), and then discharged during time-slot
t
1
(from 16:00 to 17:00, where instead the user needs more
energy than the on e allowed in low tariff). In stead, in order
to be able to stay in the low tariff area of Fig. 3 for all the
displayed 9 hours (fr om 12:00 to 21:00 ), the user flexibility
required is Q
u
= 1 kWh, with a power rate R
u
= 1 kW, since
the user is 1kW outside the low tariff area in the time-slot from
12:00 to 13:00. A possible shifted demand with respect to a (1
kWh, 1 kW) flexibility, which is always insid e the low tariff
area, is shown in the blue curve in Fig. 3. In the following, we
will call charge/discharge plan a p ower profile a
u
returnin g,
for e a ch time-slot t, the action taken on a ESS o f capacity
Q
u
and power rate R
u
. Namely, if a
t
u
≥ 0, then the ESS is
charged by a
t
u
kW (in Fig . 3 this happens, e.g., from 15:00 to
16:00). Otherwise, if a
t
u
< 0, then the ESS is discharged by
a
t
u
kW (in Fig. 3 this happens, e.g., from 12:00 to 13:00). Of
course, if a
t
u
= 0 no loads are shifted (in Fig. 3 th is happens,
e.g., f rom 13:00 to 14:00).
2) DAPP Input and Output: We now describe in deta il
input and output f or our DAPP algorith m (for a high-level
view, see Fig. 4). N a mely, DAPP req uires the following input:
1) a set of users U connected to a substation s;
2) a time-slots set T (typically with a time span one day in
the future);
3) the desire d power profile P
s
(in kW) on T for the
substation s, as decided by the DSO on the basis of EVT
output;
4) three per-unit tariffs b
l
≤ b
h
, β ∈ R
+
coming from the
energy retailer: respectively, the low (buy) price, the high
(buy) price, and the sell price for energy;
5) for each u ∈ U , a forecast d
u
for the power pro file o f
u in T (this may be compu te d on the basis of
˜
d
u
, i.e.,
of the power p rofile of u in the days preceding T using,
e.g., [28]);
6) for each user u ∈ U, th e ma ximum power (in kW) C
u
∈
R sup ported by the home main, as defined in the energy
contract for elec tricity consu mption and p roduction (e.g .,
3 or 6 kW);
7) for each user u ∈ U, the flexibility of u as a pair (R
u
, Q
u
)
(see Sect. IV-B1);
8) the minimum e nergy (in kWh) that must be contained
in the resulting low tariff are a of each user u, a s a
coefficient α multiplying the user e nergy co ntract C
u
. As
an example, if α = 2 and the maxim um power sup ported
by home u main is 6 kW, then the low tariff area of u
must contain at least 12 kWh on all period T . In our
experiments, we always u se α = 1.
The outp ut of DAPP is a set of individualised power profiles
P
u
on T , for each residential user u ∈ U . Note that each P
u
defines a low tariff a rea. Namely, the DAPP (output) tariff,
for a given user u, is defined to incentivise the u ser to follow
the output power profile region as follows: i) if u is producing
energy, then the sell price β is applied; ii) if u is consuming
energy, then either the low price b
l
or the high price b
h
are
applied, depending on the power profile of u following P
u
or
not, respectively.
3) Non-Discriminatory Price Policies: Intu itively, in order
to have residential users actually agreeing on paying bills
based on individual price policies, it is nece ssary that such
IEEE TRANSACTIONS ON SMART GRID FINAL VERSION REVISED JULY 2016 5
price policies are non-discriminatory. This is also impor tant
for the DSO an d/or th e energy retailer, as releasing discrim-
inatory price policies would decrease the number of users
accepting th e price sch ema. In our context, we informally state
that a set of individualised price policies is non- discriminatory
if and only if all residen tial users have the same opportunities
to always pay the low tariff, i.e., if individualised pr ic e
policies, in the same period of time, are all at the same relative
distance from each user’s habits. The proposed DA PP price
policies are designed to always be non-discriminatory, even in
the case that users do not perf orm load shifting in order to
follow the suggested load profile (see the discussion on the
robustness of the DAPP price policies in Section V- C).
We formally define this notion as follows. First of all, we
define a tariff system T as a triple T = (b
l
, b
h
, β, P), where
b
l
, b
h
, β ∈ R
+
are, respectively, th e low price and the high
price f or buying energy a nd the price for selling energy, as
decided by the energy retailer, whilst P = {P
u
| u ∈ U } is
the set of individualised price policies. Since pric es b
l
, b
h
, β
are equal for all residential users, they are non-discriminatory
by construction. As for P, the following holds. If the power
demand of a user u, as a response to an individualised price
policy P
u
, is
ˆ
d
t
u
on a given time-slot t, then u will pay
b
l
min{P
t
u
,
ˆ
d
t
u
}+b
h
max{
ˆ
d
t
u
−P
t
u
, 0} (here, we do not consider
the case in which
ˆ
d
t
u
< 0, as production is a lways paid with
the same price β). Since our goal is to sh ow that all users
must have the same opportunity to stay insid e the low tariff
area, we have to focus on max{
ˆ
d
t
u
− P
t
u
, 0}, wh ic h is the load
shifting w hich is required by user u in order to stay inside
the low tariff area. Intuitively, if P
u
forces user u to perform
load shifting in the same way P
u
′
does for u
′
(for any pair
u, u
′
∈ U ), then P is non-discriminatory.
In order to formally define this concept, we define, for ea ch
user u ∈ U and time-slot t ∈ T , the load shifting required b y
u in t a s:
ζ
t
u
=
max{
ˆ
d
t
u
− P
t
u
, 0}
ˆ
d
t
u
(1)
That is, the load shifting r equired by a user is the percentage
of p ower (with respect to the historical demand) which must
be shifted to keep the resulting demand inside the low tariff
area. Given the definition of required load shifting, we say that
a tariff system T is r−non-discriminatory if:
ξ = stddev
u∈U
avg
t∈T
ζ
t
u
≤ r (2)
where, for any (finite) set V , we have
avg
v∈ V
ψ(v) = µ
ψ
=
1
|V |
X
v∈ V
ψ(v) (3)
and
stddev
v∈ V
ϕ(v) =
s
1
|V |
X
v∈ V
(ϕ(v) − µ
ϕ
)
2
(4)
As an example , T is 0.1-non-discriminatory if the standard
deviation of the set of averaged load sh ifts is below 10%
(which is a reasonable thresh old in our present paper). This
Fig. 4. DAPP input and output on a single DSO substation.
allows us to measure how much the DAPP price policies
are non-discriminatory, including the case where users do not
follow the suggested power profiles (see Section V-C).
Finally, we note that estimating the flexibility of each user
is very important in order to enable DAPP to output non-
discriminatory price policies. To this aim, in our experiments,
we compute the capacity of the flexibility of ea ch residential
user u as the average on all power variations between consec-
utive time-slots in the historical power d e mand of u. As for
power rate, we fix it as the typical power ra te for ESSs, i.e.,
2kW. That is, for all u ∈ U , if the power profile of u from
the available history
˜
T is
˜
d
u
, the flexibility of u is defined as
follows:
(Q
u
, R
u
) = (τ avg
t∈
˜
T
|
˜
d
t+1
u
−
˜
d
t
u
|, 2) (5)
In this way, we assume u to be able to shift the demand of
u not more than u already does in the historical r e cords.
4) DAPP Algorith m : DAPP alg orithm consists in the fol-
lowing step s:
1) set up LP problem L;
2) solve L via an LP solver (CPLEX in our case);
3) extract the re quired output from the solu tion of L.
In the fo llowing, we describe the mathematical formulation
of the LP problem L. To this aim, first of all we list all
3|U ||T | + |T | decision variables involved.
• For each user u ∈ U and time-slot t ∈ T , a decision
variable P
t
u
, mo delling the upper bound (in kW) of the
low tariff area of user u in tim e -slot t.
• For each user u ∈ U and time-slot t ∈ T , a decision
variable b
t
u
, mod elling the state of charge (in kWh) of
the load sh ifting capability of user u in time-slot t.
• For each user u ∈ U and time-slot t ∈ T , a decision
variable a
t
u
, mo delling a charge/discharge plan, i.e., the
charge (if positive) or the discharge (if negative) action
(in kW) decided by user u in time-slot t in order to
stay within the low tariff area. Note that, in our problem
formu lation, we only consider fixed load shif ting capabil-
ities, and do not try to compute the charging/discharging
of “mobile” battery appliances such as PEVs. In our
experiments, we consider recharging of PEVs as an
additional load (that is, residential homes with a Plug-
in Electric Vehicle(s) (PEV) have higher power dem a nd).
• For each time-slot t ∈ T , a decision variable ∆
t
,
modelling the aggregated user demand (in kW) which
exceeds substation desired power profile in t.