![Fig. 4. Examples of two non-flat pricing models. The real-time prices are used by Illinois Power Company on December 15, 2009 [14]. The inclining block rates are used by British Columbia Hydro in December 2009 [19].](/figures/fig-4-examples-of-two-non-flat-pricing-models-the-real-time-2w5x15za.webp)
Optimal Residential Load Control with
Price Prediction in Real-Time
Electricity Pricing Environments
Amir-Hamed Mohsenian-Rad, Member, IEEE and Alberto Leon-Garcia, Fellow, IEEE
Abstract— Real-time electricity pricing models can potentially
lead to economic and environmental advantages compared to the
current common flat rates. In particular, they can provide end
users with the opportunity to reduce their electricity expenditures
by responding to pricing that varies with different times of
the day. However, recent studies have revealed that the lack
of knowledge among users about how to respond to time-
varying prices as well as the lack of effective building automation
systems are two major barriers for fully utilizing the potential
benefits of real-time pricing tariffs. We tackle these problems
by proposing an optimal and automatic residential energy con-
sumption scheduling framework which attempts to achieve a
desired trade-off between minimizing the electricity payment and
minimizing the waiting time for the operation of each appliance
in household in presence of a real-time pricing tariff combined
with inclining block rates. Our design requires minimum effort
from the users and is based on simple linear programming
computations. Moreover, we argue that any residential load
control strategy in real-time electricity pricing environments
requires price prediction capabilities. This is particularly true
if the utility companies provide price information only one or
two hours ahead of time. By applying a simple and efficient
weighted average price prediction filter to the actual hourly-based
price values used by the Illinois Power Company from January
2007 to December 2009, we obtain the optimal choices of the
coefficients for each day of the week to be used by the price
predictor filter. Simulation results show that the combination of
the proposed energy consumption scheduling design and the price
predictor filter leads to significant reduction not only in users’
payments but also in the resulting peak-to-average ratio in load
demand for various load scenarios. Therefore, the deployment of
the proposed optimal energy consumption scheduling schemes is
beneficial for both end users and utility companies.
Keywords: Wholesale electricity market, real-time pricing, inclin-
ing block rates, price prediction, energy consumption scheduling.
I. INTRODUCTION
Since electricity is non-storable economically, wholesale
prices (i.e., the prices set by competing generators to regional
electricity retailers) vary from day to day and usually fluctuate
by an order of magnitude between low-demand night-time
hours to high-demand afternoons. However, in general, almost
all retail consumers are currently charged some average price
that does not reflect the actual wholesale price at the time
of consumption [1]. As a remedy to this problem, various
time-differentiated pricing models have been proposed: real-
time pricing (RTP), day-ahead pricing (DAP), time-of-use
Manuscript was received on January 29, 2010; revised on May 17, 2010;
and accepted on June 7, 2010.
The authors are with the Department of Electrical and Computer Engi-
neering, University of Toronto, Toronto, ON, Canada, M5S 2E4, e-mails:
hamed@comm.utoronto.ca and alberto.leongarcia@utoronto.ca.
pricing (TOUP), critical-peak pricing (CPP), etc. In all of
these variations, the main idea is two-fold: First, allowing
retail prices to reflect fluctuating wholesale prices to the end
users so that they pay what the electricity is worth at different
times of the day; Second, encouraging users to shift high-load
household appliances to off-peak hours to not only reduce their
electricity costs but also to help to reduce the peak-to-average
ratio (PAR) in load demand
1
[2]–[5].
The research literature includes a wide range of work related
to RTP. The earliest peak-load pricing discussion dates more
than half a century ago [6], [7]. More recent theoretical and
simulation studies in [8]–[12] have focused on understanding
the economic advantages of RTP. Many of these (e.g., [10],
[11]) have proposed carefully designed tariff models in order
to improve system performance and users’ participation in
RTP and CPP programs. On the other hand, the environmental
implications of RTP are examined in [13] and it is shown
that RTP can potentially reduce the emission levels of SO
2
,
NO
x
, and CO
2
in many regions in the U.S., where peak
demand is met more by oil-fired capacity than by hydropower.
Time-differentiated pricing is currently implemented in various
regions in North America, e.g., in form of hourly-based DAP
tariff used by the Illinois Power Company in the U.S. [14]
and the three-level (on-peak, mid-peak, off-peak) TOUP tariff
used by the Ontario Hydro Company in Toronto, Canada [15].
Another alternative to the common flat rates in retail elec-
tricity market is the conservation rates model with inclining
block rates (IBR). In IBR pricing, the marginal price increases
by the total quantity consumed [16]. That is, beyond a certain
threshold in the total monthly/daily/hourly residential load, the
electricity price will increase to a higher value. This creates
incentives for end users to conserve to distribute their load
at different times of the day in order to avoid paying for
electricity at higher rates. In addition, IBR helps in load
balancing and reducing the PAR [17]. It has been widely
adopted in the pricing tariffs by some utility companies since
the 1980s. For example, the Southern California Edison, San
Diego Gas & Electric, and Pacific Gas & Electric companies
currently have two-level residential rate structures where the
marginal price in the second level (i.e., the higher block)
is 80% or higher than the first level (i.e., the lower block),
depending on the utility [18]. In Canada, the British Columbia
1
Appropriate load-shifting is foreseen to become even more crucial as plug-
in hybrid electric vehicles (PHEVs) become more popular. Most PHEVs need
0.2 - 0.3 kWh of charging power for one mile of driving [2]. This will
represent a major new load on the existing distribution system. In particular,
during the charging time, the PHEVs double the average household load [2].
Hydro Company currently uses a two-level conservation rate
structure with 40% higher prices at the second level [19].
Recent studies have shown that despite several advantages
that RTP and IBR can offer, the lack of knowledge among the
users about how to respond to time-varying prices and the lack
of effective home automation systems are two major barriers
for fully utilizing the benefits of real-time pricing tariffs [20],
[21]. In fact, most of the current residential load control
activities are operated manually. This makes it difficult for
users to optimally schedule the operation of their appliances in
response to the hourly updated pricing information they may
receive from the utilities in an RTP program. For example,
the experience of the RTP program in Chicago has shown that
although the price values were available via telephone and
the Internet, only rarely did households actively check prices
as it was difficult for the participants to constantly monitor
the hourly prices to respond properly [1]. Another example is
the results from a more recent study by The Utility Reform
Network (TURN) in San Francisco which has reported that
most users do not have time and knowledge to even pursue
their own interest while they respond to real-time prices [22].
The main focus of this paper is on proposing a computa-
tionally feasible and automated optimization-based residential
load control scheme in a retail electricity market with RTP
combined with IBR. We aim to minimize the household’s
electricity payment by optimally scheduling the operation and
energy consumption for each appliance, subject to the special
needs indicated by the users. We assume that each residential
consumer is equipped with a smart meter [23], connected
to a smart power distribution system with a two-way digital
communication capability through computer networking [24]–
[26]. While periodically receiving the updated information
on prices from the utility, each smart meter includes an
energy scheduling unit which decides on energy consumption
in the household. Depending on the scheduling horizon, the
operation of the energy scheduling unit is complemented by a
price predictor unit which estimates the upcoming prices by
applying a weighted averaging filter to past prices. We obtain
the optimal coefficients of the price predictor filter and show
that it is best to use different coefficients at different days of
the week. In this regard, we use the actual RTP tariffs adopted
by the Illinois Power Company (IPC) from January 2007 to
December 2009 which was available to public online at [27].
The results and analysis in this paper differ from the related
work in the literature in several aspects. Unlike [9] we do
not focus on understanding the residential user’s response to
RTP models. Instead, we try to help the users to shape their
response properly and in an automated fashion. Our work is
also different from the heuristic home automation schemes
in [21], [28] as here we use an optimization-based approach
with elaborate mathematical analysis. Furthermore, the IP-
based networking architecture proposed for home automation
in [21] can also be used for the implementation of our design
in practice. While the optimization problem we study in this
paper is partly similar to the one studied in [17] for fixed
prices, here we take into account time-varying prices as well
as the trade-off between minimizing the electricity payment
and minimizing the waiting time for the operation of each
appliance. In addition, the optimization problem we study here
is more realistic with respect to price models but requires
more efforts to be solved due to the non-differentiability of the
objective function. Last but not least, price prediction is not
studied in [9], [17]. In fact, to the best of our knowledge, none
of the prior work on residential load control has considered
real-time price prediction at the user side.
The rest of this paper is organized as follows. We introduce
the system model and notation in Section II. In Section III,
we discuss the price prediction problem and introduce our
weighted average price prediction filter which is designed and
evaluated on a weekly basis, using the actual hourly price
values adopted by the IPC. Our proposed linear programming
scheme for optimal load control is introduced in Section IV.
Remarks, special cases, and extensions are highlighted in
Section V. Simulation results are provided and discussed in
Section VI. The paper is concluded in Section VII.
II. SYSTEM MODEL
In this section, we provide a mathematical representation
of the residential load control problem in RTP environments
with IBR. We consider the general wholesale electricity market
scenario shown in Fig. 1, where each retailer/utility serves
a number of end users. The RTP information, reflecting the
wholesale prices, are informed by the retailer to the users
over a digital communication infrastructure, e.g., a local area
network (LAN). In this scenario, our focus is to formulate the
energy consumption scheduling problem in each household
as an optimization problem that aims to achieve a trade-off
between minimizing the electricity payment and minimizing
the waiting time for the operation of each household appliance
in response to the real-time prices announced by the retailer
company. We will explain how the optimization problem in
this section can be solved in practice later in Section IV.
A. Residential Consumers
Consider a residential unit that participates in a real-time
pricing program. Let A denote the set of appliances in this unit
which may include washer/dryer, refrigerator, plug-in hybrid
vehicle, etc. For each appliance a ∈ A, we define an energy
consumption scheduling vector x
a
as follows:
x
a
, [x
1
a
, . . . , x
H
a
], (1)
where H ≥ 1 is the scheduling horizon that indicates the
number of hours ahead which are taken into account for deci-
sion making in energy consumption scheduling. For example,
H = 24 or H = 48. For each upcoming hour of the day
h ∈ H , {1, . . . , H}, a real-valued scalar x
h
a
≥ 0 denotes the
corresponding one-hour energy consumption that is scheduled
for appliance a ∈ A. On the other hand, let E
a
denote the
total energy needed for the operation of appliance a ∈ A.
For example, in case of a plug-in hybrid electric sedan, in
total E
a
= 16 kWh is needed to charge the battery for a
40-miles driving range [2]. As another example, for a typical
front-loading clothes washing machine with warm wash/rinse
setting, we have E
a
= 3.6 kWh per load [29]. Next, assume
that for each appliance a ∈ A, the user indicates α
a
, β
a
∈ H
Fig. 1. A simplified illustration of the wholesale electricity market formed by multiple generators and several regional retail companies. Each retailer provides
electricity for a number of users. Retailers are connected to the users via local area networks which are used to announce real-time prices to the users.
as the beginning and end of a time interval in which the
energy consumption for appliance a is valid to be scheduled,
respectively. Clearly, we always have α
a
< β
a
. For example,
after loading a dishwasher with the dishes used at the lunch
table, the user may select α
a
= 2 PM and β
a
= 6 PM for
scheduling the energy consumption for the dishwasher as he
expects the dishes to be ready to use by dinner time in the
evening. As another example, the user may select α
a
= 10 PM
and β
a
= 7 AM (the next day) for his PHEV after plugging
it in at night such that the battery charging finishes by early
morning time when he needs to use the vehicle to go to work.
Given the pre-determined parameters E
a
, α
a
, and β
a
, in order
to provide the needed energy for each appliance a ∈ A in
times within the interval [α
a
, β
a
], it is required that
β
a
h=α
a
x
h
a
= E
a
. (2)
Further to constraint (2), it is expected that x
a
= 0 for any
h < α
a
and h > β
a
as no operation (thus energy consumption)
is needed outside the time frame [α
a
, β
a
] for appliance a. We
note that the time length β
a
− α
a
needs to be larger than
or equal to the time duration required to finish the normal
operation of appliance a. For example, for a single-phase
PHEV, the normal charging time is 3 hours [2]. Therefore,
it is required that β
a
− α
a
≥ 3. Clearly, if β
a
− α
a
= 3 the
timing imposed by the user would be strict and any energy
consumption scheduling strategy has no choice but arranging
full power charging within the whole interval [α
a
, β
a
]. On
the other hand, if β
a
− α
a
≫ 3, it is possible to select
certain hours within the large interval [α
a
, β
a
] to schedule
energy consumption such that the electricity payments can be
minimized. We will further discuss this issue in Section II-C.
All home appliances have certain maximum power levels
denoted by γ
max
a
, for each a ∈ A. For example, a PHEV
may be charged only up to γ
max
a
= 3.3 kW per hour [2].
Some appliances may also have minimum stand-by power
levels γ
min
a
, for each a ∈ A. Therefore, the following lower
and upper bound constraints are needed on the choices of the
energy scheduling vector x
a
for each appliance a ∈ A:
γ
min
a
≤ x
h
a
≤ γ
max
a
, ∀ h ∈ [α
a
, β
a
]. (3)
Finally, we note that there is usually a limit on the total
energy consumption at each residential unit at each hour. This
limit, denoted by E
max
, can be set by the utility to impose
the following set of constraints on energy scheduling:
a∈A
x
h
a
≤ E
max
, ∀ h ∈ H. (4)
Together, constraints (2)-(4) determine all valid choices for
the energy consumption scheduling vectors. Therefore, we can
define a feasible scheduling set X for all possible energy
consumption scheduling vectors as
X =
x |
β
a
h=α
a
x
h
a
= E
a
, ∀ a ∈ A,
γ
min
a
≤ x
h
a
≤ γ
max
a
, ∀ a ∈ A, h ∈ [α
a
, β
a
],
x
h
a
= 0, ∀ a ∈ A, h ∈ H\[α
a
, β
a
],
a∈A
x
h
a
≤ E
max
, ∀ h ∈ H
,
(5)
where x , (x
a
, ∀a ∈ A) denotes the vector of energy con-
sumption scheduling variables for all appliances. An energy
schedule x is valid only if x ∈ X . Clearly, the proper choice
of x would depend on the electricity prices. In this regard,
we assume that each household is equipped with a smart
meter as shown in Fig. 2. The real-time prices are provided
by the utility company via a LAN. The user announces his
needs by selecting parameters E
a
, α
a
, β
a
, γ
min
a
, and γ
max
a
for each appliance a ∈ A. Then, the energy scheduler, with
some help form the price predictor if needed, determines the
optimal choice of energy consumption scheduling vector x.
The resulting energy consumption schedule is then applied to
all household appliances in form of on/off commands with
specified power levels over a wired or wireless home area
network among the appliances and the smart meter. An exam-
ple wireless home area network (WHAN) is shown in Fig. 3.
In this setting, the in-home wireless communications can be
implemented by ZigBee transceivers, offered by the ZigBee
Alliance [30]. Another candidate for in-home communication
is HomePlug power-line communication technology, offered
by HomePlug Powerline Alliance [31]. More details about
various home area network technologies can be found in [32].
Next, we discuss the details on the real-time pricing model
as well as our proposed optimization-based load control strat-
egy in Sections II-B and II-C, respectively.
LAN
Power Line
Appliances
Price Predictor
Energy Scheduller
Electricity
Real-Time Prices
p
h
p
h
(l )
h
(l )
h
l
h
Smart Meter
x
User’s
Needs
Fig. 2. The operation of smart meter in our design. Given the real-time
prices p
h
(l
h
) for all h ∈ P from the utility, there are two main units involved
in residential load control: energy scheduler and price predictor. The latter
estimates the upcoming prices which are not announced by the utility, i.e.,
ˆp
h
(l
h
) for all h ∈ H\P if the price announcement horizon P is less that the
scheduling horizon H. The proper choices of energy consumption scheduling
vectors x
a
for all appliances a ∈ A is determined by the energy scheduler
unit based on the solution of optimization problem (25).
B. Real-Time Pricing with Inclining Block Rates
Recall from Section I that RTP and IBR are two promising
non-flat pricing models to replace the current flat rate tariffs.
In this section, we provide a general mathematical pricing
representation which combines these two pricing models. For
now, we assume that the future pricing parameters are known
for the users ahead of time. This is indeed the case in DAP
structures. We will discuss price prediction in Section III.
Let l
h
,
a∈A
x
h
a
denote the total hourly household
energy consumption at each upcoming hour h ∈ H. Recall
that H denotes the scheduling horizon. We consider a gen-
eral hourly pricing function p
h
(l
h
) which depends on three
parameters a
h
, b
h
, c
h
≥ 0 and is formulated as follows:
p
h
(l
h
) =
a
h
, if 0 ≤ l
h
≤ c
h
,
b
h
, if l
h
> c
h
.
(6)
It is clear that the price model in (6) is not a flat rate structure
as the price value depends on time of day and total load.
In fact, the price model in (6) represents an RTP structure
combined with IBR. To see this, let us consider the example
pricing models shown in Figs. 1(a) and (b) which are currently
implemented by IPC in the U.S. and British Columbia Hydro
Company in Canada, respectively. These examples are both
special cases of the more general pricing model in (6). For
the RTP model used by IPC in Fig. 1(a), we have
a
h
= b
h
, ∀ h ∈ H. (7)
That is, although the prices vary every hour, they are indeed
flat within each hour. On the other hand, for the IBR used by
British Columbia Hydro Company, we have
a
1
= a
2
= . . . = a
H−1
= a
H
, (8)
b
1
= b
2
= . . . = b
H−1
= b
H
, (9)
c
1
= c
2
= . . . = c
H−1
= c
H
. (10)
That is, although the prices are dependent on consumption
level, they do not change over time; thus, they cannot reflect
the fluctuations in the wholesale prices. By combining the
two pricing scenarios in (6), both wholesale prices as well
as consumption levels are taken into account.
Fig. 3. The resulting energy consumption schedules selected by the energy
scheduler can be applied to household appliances in form of operation
commands over a wireless home area network using ZigBee transceivers.
C. Problem Formulation
Given the feasible energy scheduling set X and the RTP
model in (6), the key question is: How should each user’s
energy consumption be scheduled in response to time-varying
prices? Before answering this question, we first argue that
the user’s interest is two fold. First, each user wishes to
minimize his payment. In fact, it is reasonable to assume
that all users care about the amount on their electricity bills.
Second, depending on the appliance, some users may also
care about their comfort and getting the work done (e.g.,
washing their dishes, charging their PHEV, or cleaning their
clothes) as soon as possible. Clearly, these two objectives can
be conflicting in many scenarios. For example, in case of the
RTP structure in Fig. 4(a) and when the user wants to start
washing dishes at 9:00 AM right after finishing the breakfast,
he may choose to wait for 5 hours and postpone the operation
of the dishwasher (with E
a
= 3.6 kWh per load) to 2:00 PM
in order to reduce the corresponding electricity payment from
3.6 × 4.1 = 14.8 cents to 3.6 × 2.9 = 10.6 cents and save 4.2
cents. However, for some reason, the user may prefer to pay
the extra 4.2 cents and finish the work by 10:00 AM. As an
alternative, the user might be willing to wait for 2 hours only
and save 1.5 cents instead. In fact, we can see that there is
a trade-off involved between the two design objectives. Next,
we explain how this trade-off can be mathematically taken into
account in an optimization-based framework.
From the RTP model introduced in Section II-B, the user’s
total electricity payment corresponding to all appliances within
the upcoming scheduling horizon is obtained as
H
h=1
p
h
a∈A
x
h
a
×
a∈A
x
h
a
, (11)
where the price function p
h
(·) is as in (6). On the other hand,
the cost of waiting can be modeled as
H
h=1
a∈A
ρ
h
a
x
h
a
. (12)
Here, for each appliance a ∈ A and any hour h ∈ H, the
waiting parameter ρ
h
a
≥ 0. Clearly, ρ
h
a
= 0 for all h < α
a
and
h > β
a
as the concept of waiting may only be defined within
the valid scheduling interval [α
a
, β
a
]. On the other hand, it is
reasonable to assume that we always have
ρ
α
a
a
≤ . . . ≤ ρ
β
a
a
, ∀ a ∈ A. (13)
0 2 4 6 8 10 12 14 16 18 20 22 24
0
1
2
3
4
5
6
7
8
9
Time of Day (Hour) h
p
h
(l
h
) (Cents/kWh)
Real−time Pricing
Peak Hours
(a) Real-time prices set by Illinois Power Company.
0 0.5 1 1.5 2
3
4
5
6
7
8
9
10
11
12
Residential Load l
h
p
h
(l
h
) (Cents/kWh)
Two−level Inclining Block (Conservation) Rates
Level 1: 5.91 Cents/kWh
Level 2: 8.27 Cents/kWh
Threshold
(Base Load)
(Hihg Load)
(b) Two-level inclining block rates set by BC Hydro.
Fig. 4. Examples of two non-flat pricing models. The real-time prices are
used by Illinois Power Company on December 15, 2009 [14]. The inclining
block rates are used by British Columbia Hydro in December 2009 [19].
That is, the cost of waiting increases as more energy consump-
tion is scheduled at later hours. In particular, one can use the
following model for the waiting parameter for each a ∈ A:
ρ
h
a
=
(δ
a
)
β
a
−h
E
a
, ∀ a ∈ A, h ∈ [α
a
, β
a
], (14)
where δ
a
≥ 1 is an adjustable control parameter. The higher
the value of parameter δ
a
the higher will be the cost of waiting.
We are now ready to formulate the energy consumption
scheduling problem as the following optimization problem:
minimize
x∈X
H
h=1
p
h
a∈A
x
h
a
a∈A
x
h
a
+ λ
wait
H
h=1
a∈A
(δ
a
)
β
a
−h
x
h
a
E
a
,
(15)
where the optimization variables are the energy consumption
scheduling vectors x
a
for all appliances a ∈ A. The first
and the second terms in the objective function in (15) denote
the total electricity payment amount and the total cost of
waiting across all appliances, respectively. Here, parameter
λ
wait
is used in order to control the importance of the waiting
cost terms in the objective function of the proposed design
optimization problem. A typical value for this parameter can
be λ
wait
= 1. On the other hand, parameter δ
a
acts as a knob to
control the trade-off between the two design objectives with
respect to minimizing the payment and the waiting cost for
each appliance. Clearly, a user may assign different values δ
a
for different appliances. As a special case, we notice that if
for a certain appliance a ∈ A we have δ
a
= 1, then
H
h=1
(δ
a
)
β
a
−h
x
h
a
E
a
= 1, ∀ x ∈ X , (16)
where the equality is due to (2). Therefore, the waiting cost
will have no impact on the solution of optimization problem
(15). In practice, three choices of parameter δ
a
can be pre-
determined and labeled as three operation modes:
• Strict Cost Reduction: δ
a
= 1,
• Medium Cost Reduction: δ > 1,
• No Cost Reduction: δ
a
≫ 1.
Next, we address the issue of price prediction. Then, we will
explain how to solve problem (15) in practice with low com-
putational complexity in Section IV. Note that optimization
problem (15) is not tractable in its current form due to the
non-differentiability of the price function p
h
(l
h
) in (6).
III. PRICE PREDICTION IN REAL-TIME PRICING
ELECTRICITY ENVIRONMENTS
So far, we have assumed that each end user is fully aware
of the upcoming price values set by the utility company within
the scheduling horizon H. That is, the user always knows the
values of a
h
, b
h
, and c
h
for each h ∈ H. This assumption can
be valid in certain practical scenarios such as in DAP where
the utility company releases the pricing details for the next
24 hours on a daily basis. Examples of such real-time pricing
tariffs include the one implemented by IPC [14]. However,
we may consider more dynamic pricing scenarios where the
upcoming prices are announced only for 1 ≤ P ≪ H
hours ahead of time. Here, P denotes the price announcement
horizon. For example, we may have P = 5 or 6 hours. Clearly,
the extreme case would be P = 1 when only the next hour
price is released. In these cases, any energy consumption
scheduling policy, including the optimization-based energy
consumption scheduling approach described in Section II-C,
essentially requires some price prediction capabilities.
A. Prediction Based on Prior Knowledge
In general, price parameters may depend on several factors.
In particular, they depend on the wholesale market prices
which are not easy to predict themselves. Nevertheless, it
is usually expected that the prices are higher during the
afternoon, on hot days in the summer, and on cold days in
the winter [1]. Furthermore, one may expect that the prices
vary depending on the working days or weekends. These
pieces of information can potentially help in predicting the