# Matching of Everyday Power Supply and Demand with Dynamic Pricing:
Problem Formalisation and Conceptual Analysis

Thibaut Théate<EMAIL_ADDRESS>Antonio Sutera  Damien Ernst
<EMAIL_ADDRESS>Department of Electrical Engineering and Computer Science,
University of Liège, Liège, Belgium  Haulogy, Intelligent Systems Solutions,
Braine-le-Comte, Belgium  Information Processing and Communications
Laboratory, Institut Polytechnique de Paris, Paris, France

###### Abstract

The energy transition is expected to significantly increase the share of
renewable energy sources whose production is intermittent in the electricity
mix. Apart from key benefits, this development has the major drawback of
generating a mismatch between power supply and demand. The innovative dynamic
pricing approach may significantly contribute to mitigating that critical
problem by taking advantage of the flexibility offered by the demand side. At
its core, this idea consists in providing the consumer with a price signal
which is evolving over time, in order to influence its consumption. This novel
approach involves a challenging decision-making problem that can be summarised
as follows: how to determine a price signal maximising the synchronisation
between power supply and demand under the constraints of maintaining the
producer/retailer’s profitability and benefiting the final consumer at the
same time? As a contribution, this research work presents a detailed
formalisation of this particular decision-making problem. Moreover, the paper
discusses the diverse algorithmic components required to efficiently design a
dynamic pricing policy: different forecasting models together with an accurate
statistical modelling of the demand response to dynamic prices.

###### keywords:

Matching of supply and demand , dynamic pricing , demand response , power
producer/retailer.

##  1 Introduction

Climate change is undeniably a major challenge facing humanity in the 21st
century [1]. An ambitious transformation is required in all sectors to
significantly lower their respective carbon footprints. Electricity generation
is no exception, with the burning of fossil fuels, mainly coal and gas, being
by far the dominant power source in the world today [2]. This sector has to
undergo an important transformation of the global electricity mix by promoting
power sources with a significantly lower carbon footprint. Belonging to that
category are nuclear power, hydroelectricity, biomass or geothermal energy
which are relatively controllable, but also the energy directly extracted from
wind and sun which is conversely intermittent in nature. Since wind turbines
and photovoltaic panels are expected to play a key role in the energy
transition, solutions are required to address their variable production.
Interesting technical avenues are the interconnection of power grids [3] and
the development of storage capacities such as battery, pumped hydroelectricity
or hydrogen [4]. Another promising and innovative solution is to influence the
behaviour of consumers through the use of dynamic pricing (DP), so that power
supply and demand are better synchronised.

The dynamic pricing approach consists in continuously adapting the electricity
price that the final consumer has to pay in order to influence its consumption
behaviour. Basically, when demand exceeds supply, the power price would be
increased in order to take down consumption. Conversely, a reduced price would
be provided when there is excessive production compared to consumption. From a
graphical perspective, the objective is not only to shift the daily
consumption curve but also to change its shape in order to better overlap with
the intermittent production curve of renewable energy sources. This is
illustrated in Figure 1 for a representative situation.

Figure 1: Illustration of the dynamic pricing approach’s potential to shift
and change the shape of a typical daily consumption curve (blue) so that there
is a better synchronisation with the daily intermittent production curve of
renewable energy sources (red).

The innovative dynamic pricing approach relies on two important assumptions.
Firstly, the final consumer has to be equipped with a smart metering device to
measure its production in real-time and with communication means for the price
signal. Secondly, the final consumer has to be able to provide a certain
amount of flexibility regarding its power consumption. Moreover, it has to be
sufficiently receptive to the incentives offered to reduce its electricity
bill in exchange for a behaviour change. If these requirements are met, the
major strength of the dynamic pricing approach is its potential benefits for
both the consumer and the producer/retailer. Moreover, these benefits would
not only be in terms of economy, but also potentially in terms of ecology and
autonomy. In fact, dynamic prices reward the flexibility of the demand side.

The contributions of this research work are twofold. Firstly, the complex
decision-making problem faced by a producer/retailer willing to develop a
dynamic pricing strategy is presented and rigorously formalised. Secondly, the
diverse algorithmic components required to efficiently design a dynamic
pricing policy are thoroughly discussed. To the authors’ knowledge, demand
response via dynamic pricing has received considerable attention from the
research community, but from the perspective of the demand side alone.
Therefore, the present research may be considered as a pioneer work studying
dynamic pricing from the perspective of the supply side for taking advantage
of the flexibility of the power consumers.

This research paper is structured as follows. First of all, the scientific
literature about both dynamic pricing and demand response is concisely
reviewed in Section 2. Then, Section 3 presents a detailed formalisation of
the decision-making problem behind the novel dynamic pricing approach.
Afterwards, Section 4 analyses the algorithmic components necessary for the
development of dynamic pricing policies. Subsequently, a fair performance
assessment methodology is introduced in Section 5 to quantitatively evaluate
the performance of a dynamic pricing policy. To end this paper, Section 6
discusses interesting research avenues for future work and draws conclusions.

##  2 Literature review

Over the last decade, the management of the demand side in the scope of the
energy transition has received increasing attention from the research
community. In fact, there exist multiple generic approaches when it comes to
demand response. Without getting into too many details, the scientific
literature includes some surveys summarising and discussing the different
techniques available together with their associated challenges and benefits
[5, 6, 7, 8, 9]. In this research work, the focus is exclusively set on the
demand response induced by dynamic power prices.

As previously mentioned, the scientific literature about demand response via
dynamic pricing is primarily focused on the perspective of the demand side.
Multiple techniques have already been proposed to help the consumer provide
flexibility and take advantage of behavioural changes to lower its electricity
bill. For instance, [10] presents a power scheduling method based on a genetic
algorithm to optimise residential demand response via an energy management
system, so that the electricity cost is reduced. In [11], a technique based on
dynamic programming is introduced for determining the optimal schedule of
residential controllable appliances in the context of time-varying power
pricing. One can also mention [12] that proposes an energy sharing model with
price-based demand response for microgrids of peer-to-peer prosumers. The
approach is based on a distributed iterative algorithm and has been shown to
lower the prosumers’ costs and improve the sharing of photovoltaic energy.
More recently, (deep) reinforcement learning techniques have been proven to be
particularly relevant for controlling the residential demand response in the
context of dynamic power prices [13, 14].

On the contrary, the question of inducing a residential demand response based
on a dynamic pricing approach from the perspective of the supply side has not
received a lot of attention from the research community yet. Still, there are
a few works in the scientific literature about the mathematical modelling of
the demand response caused by dynamic power prices, which is a key element in
achieving that objective. To begin with, [15] presents a simulation model
highlighting the evolution of electricity consumption profiles when shifting
from a fixed tariff to dynamic power prices. The same objective is pursued by
[16] which introduces a fully data-driven approach relying on the data
collected by smart meters and exogenous variables. The resulting simulation
model is based on consumption profiles clustering and conditional variational
autoencoders. Alternatively, [17] presents a functional model of residential
power consumption elasticity under dynamic pricing to assess the impact of
different electricity price levels, based on a Bayesian probabilistic
approach. In addition to these mathematical models, one can also mention some
real-life experiments conducted to assess the responsiveness of residential
electricity demand to dynamic pricing [18, 19].

##  3 Problem formalisation

This section presents a mathematical formalisation of the challenging
sequential decision-making problem related to the dynamic pricing approach for
inducing a residential demand response. To begin with, the contextualisation
considered for studying this particular problem is briefly described, followed
by an overview of the decision-making process. Then, a discretisation of the
continuous timeline is introduced. Subsequently, the formal definition of a
dynamic pricing policy is presented. Lastly, the input and output spaces of a
dynamic pricing policy are described, together with the objective criterion.

###  3.1 Contextualisation

As previously hinted, this research work focuses on the interesting real-case
scenario of a producer/retailer whose production portfolio is composed of an
important share of renewable energy sources such as wind turbines and
photovoltaic panels. Because of the substantial intermittency of these
generation assets, a strong connection to the energy markets is required in
order to fully satisfy its customers regardless of the weather. Nevertheless,
the consumers are assumed to be well informed and willing to adapt their
behaviour in order to consume renewable energy rather than electricity
purchased on the market whose origin may be unknown. Within this particular
context, the benefits of the dynamic pricing approach taking advantage of the
consumers’ flexibility are maximised. Indeed, the insignificant marginal cost
associated with these intermittent renewable energy sources coupled with their
low carbon footprint make this innovative approach interesting from an
economical perspective for both supply and demand sides, but also in terms of
ecology. Moreover, the autonomy of the producer/retailer is expected to be
reinforced by lowering its dependence on the energy markets. At the same time,
dependence on fossil fuels may be reduced as well.

In this research work, the predicted difference between power production and
consumption is assumed to be fully secured in the day-ahead electricity
market. Also called spot market, the day-ahead market has an hourly resolution
and is operated once a day for all hours of the following day via a single-
blind auction. In other words, trading power for hour $H$ of day $D$ has to be
performed ahead on day $D-1$ between 00:00 AM (market opening) and 12:00 AM
(market closure). Therefore, the energy is at best purchased 12 hours (00:00
AM of day $D$) up to 35 hours (11:00 PM of day $D$) before the actual delivery
of power. Apart from the day-ahead electricity market, it is assumed that
there are no trading activities on the future/forward nor intraday markets.
Nevertheless, if there remains an eventual mismatch between production and
consumption at the time of power delivery, the producer/retailer would be
exposed to the imbalance market. In this case, the so-called imbalance price
has to be inevitably paid as compensation for pushing the power grid off
balance.

###  3.2 Decision-making process overview

The decision-making problem studied in this research work is characterised by
a particularity: a variable time lag between the moment a decision is made and
the moment it becomes effective. As previously explained, any remaining
difference between production and consumption after demand response has to
ideally be traded on the day-ahead market. The purpose of this assumption is
to limit the exposure of the producer/retailer to the imbalance market. For
this reason, the price signal sent to the consumer on day $D$ has to be
generated before the closing of the day-ahead market on day $D-1$.
Additionally, it is assumed that the price signal cannot be refreshed
afterwards.

Basically, the decision-making problem at hand can be formalised as follows.
The core objective is to determine a decision-making policy, denoted $\Pi$,
mapping at time $\tau$ input information of diverse nature $I_{\tau}$ to the
electricity price signal $S_{\tau}$ to be sent to the consumers over a future
time horizon well-defined:

$S_{\tau}=\Pi(I_{\tau})\text{,}$ (1)

where:

  * $\bullet$

$I_{\tau}$ represents the information vector gathering all the available
information (of diverse nature) at time $\tau$ which may be helpful to make a
relevant dynamic pricing decision,

  * $\bullet$

$S_{\tau}$ represents a set of electricity prices generated at time $\tau$ and
shaping the dynamic price signal over a well-defined future time horizon.

The dynamic pricing approach from the perspective of the supply side belongs
to a particular class of decision-making problems: automated planning and
scheduling. Contrarily to conventional decision-making outputting one action
at a time, planning decision-making is concerned with the generation of a
sequence of actions. In other words, a planning decision-making problem
requires to synthesise in advance a strategy or plan of actions over a certain
time horizon. Formally, the decision-making has to be performed at a specific
time $\tau$ about a control variable over a future time horizon beginning at
time $\tau_{i}>\tau$ and ending at time $\tau_{f}>\tau_{i}$. In this case, the
decision-making is assumed to be performed just before the closing of the day-
ahead market at 12:00 AM to determine the price signal to be sent to the
consumers throughout the entire following day (from 00:00 AM to 11:59 PM).

In the next sections, a more accurate and thorough mathematical formalisation
of the dynamic pricing problem from the perspective of the supply side is
presented. Moreover, the planning problem previously introduced is cast into a
sequential decision-making problem. Indeed, this research paper intends to
focus on a decision-making policy outputting a single price from the signal
$S_{\tau}$ at a time based on a subset of the information vector $I_{\tau}$.

Figure 2: Illustration of the formalised decision-making problem related to
dynamic pricing from the perspective of the supply side. The notations $x_{t}$
and $y_{t}$ represent the inputs and outputs of a dynamic pricing policy
$\pi$, which are not shown concurrent on the timeline since the decision-
making occurs multiple hours before the application of the dynamic pricing
signal. The time axis of the four plots represents the complete following day
for which the dynamic prices are generated. The mathematical notations
$p_{t}^{F}$, $c_{t}^{F}$ and $\lambda_{t}^{F}$ respectively represent the
forecast production, consumption and day-ahead market price for the time step
$t$. Finally, the quantity $c_{t}^{\prime}$ is the predicted consumption at
time step $t$ after taking into consideration the dynamic pricing signal.

###  3.3 Timeline discretisation

Theoretically, the dynamic electricity price signal sent to the consumer could
be continuously changing over time. More realistically, this research work
adopts a discretisation of the continuous timeline so that this power price is
adapted at regular intervals. Formally, this timeline is discretised into a
number of time steps $t$ spaced by a constant duration $\Delta t$. If the
duration $\Delta t$ is too large, the synchronisation improvement between
supply and demand will probably be of poor quality. Conversely, lowering the
value of the duration $\Delta t$ increases the complexity of the decision-
making process, and a too high update frequency may even confuse the consumer.
There is a trade-off to be found concerning this important parameter. In this
research work, the dynamic price signal is assumed to change once per hour,
meaning that $\Delta t$ is equal to one hour. This choice is motivated by the
hourly resolution of the day-ahead market, which has proven to be an
appropriate compromise over the years for matching power production and
consumption. Another relevant discretisation choice could be to have a price
signal which is updated every quarter of an hour. In the rest of this research
paper, the increment (decrement) operations $t+1$ ($t-1$) are used to model
the discrete transition from time step t to time step $t+\Delta t$ ($t-\Delta
t$), for the sake of clarity.

###  3.4 Dynamic pricing policy

Within the context previously described, a dynamic pricing planning policy
$\Pi$ consists of the set of rules used to make a decision regarding the
future price signal sent to the consumers over the next day. This planning
policy can be decomposed into a set of 24 dynamic pricing decision-making
policies $\pi$ outputting a single electricity price for one hour of the
following day. Mathematically, such a dynamic pricing strategy can be defined
as a programmed policy $\pi:\mathcal{X}\rightarrow\mathcal{Y}$, either
deterministic or stochastic, which outputs a decision $y_{t}\in Y$ for time
step $t$ based on some input information $x_{t}\in\mathcal{X}$ so as to
maximise an objective criterion. The input $x_{t}$ is derived from the
information vector $I_{\tau}$ associated with the decision-making for time
step $t$, after potential preprocessing operations. The price signal
$S_{\tau}$ is composed of 24 dynamic pricing policy outputs $y_{t}$.

In the rest of this research work, the time at which the decision-making does
occur should not be confused with the time at which the dynamic price signal
is active (charging for energy consumption). The proposed formalisation
assumes that the time step $t$ refers to the time at which the dynamic price
is active, not decided. Therefore, the decision-making of the dynamic pricing
policy for time step $t$ ($y_{t}=\pi(x_{t})$) is in fact performed hours in
advance of time step $t$. This complexity is illustrated in Figure 2
describing the formalised decision-making problem.

###  3.5 Input of a dynamic pricing policy

The input space $\mathcal{X}$ of a dynamic pricing policy $\pi$ comprises all
the available information which may help to make a relevant decision about
future electricity prices so that an appropriate demand response is induced.
Since the decision-making occurs 12 up to 35 hours in advance of the price
signal delivery, this information mainly consists of forecasts and estimations
that are subject to uncertainty. As depicted in Figure 2, the dynamic pricing
policy input $x_{t}\in\mathcal{X}$ refers to the decision-making occurring at
time $\tau=t-h$ with $h\in[12,\ 35]$ about the dynamic pricing signal
delivered to the consumer at time step $t$. In fact, the quantity $I_{\tau}$
may be seen as the information contained in the 24 inputs $x_{t}$ for
$t\in\\{\tau+12,...,\tau+35\\}$. Formally, the input $x_{t}\in\mathcal{X}$ is
decided to be defined as follows:

$x_{t}=\\{P_{t}^{F},\ C_{t}^{F},\ \Lambda_{t}^{F},\ Y_{t},\
\mathcal{M}\\}\text{,}$ (2)

where:

  * $\bullet$

$P_{t}^{F}=\\{p_{t+\epsilon}^{F}\in\mathbb{R}^{+}\ |\ \epsilon=-k,...,k\\}$
represents a set of forecasts for the power production within a time window
centred around time step $t$ and of size $k$,

  * $\bullet$

$C_{t}^{F}=\\{c_{t+\epsilon}^{F}\in\mathbb{R}^{+}\ |\ \epsilon=-k,...,k\\}$
represents a set of forecasts for the power consumption within a time window
centred around time step $t$ and of size $k$,

  * $\bullet$

$\Lambda_{t}^{F}=\\{\lambda_{t+\epsilon}^{F}\in\mathbb{R}\ |\
\epsilon=-k,...,k\\}$ represents a set of forecasts for the day-ahead market
prices within a window centred around time step $t$ and of size $k$,

  * $\bullet$

$Y_{t}=\\{y_{t-\epsilon}\in\mathbb{R}\ |\ \epsilon=1,...,k\\}$ represents the
series of $k$ previous values for the dynamic price signal sent to the final
consumer,

  * $\bullet$

$\mathcal{M}$ is a mathematical model of the demand response to be expected
from the consumption portfolio, with the required input information.

The different forecasting models and the challenging modelling of the
consumption portfolio demand response are discussed in more details in Section
4.

###  3.6 Output of a dynamic pricing policy

The output space $\mathcal{Y}$ of a dynamic pricing policy $\pi$ only includes
the future price signal to be sent to the consumer. Formally, the dynamic
pricing policy output $y_{t}\in\mathcal{Y}$, which represents the electricity
price to be paid by the consumer for its power consumption at time step $t$,
is mathematically defined as follows:

$y_{t}=e_{t}\text{,}$ (3)

where $e_{t}\in\mathbb{R}$ represents the dynamic electricity price to be paid
by the demand side for its power consumption at time step $t$. Out of the
scope of this research work is the presentation of this price signal so that
the impact on the final consumer is maximised. Indeed, the way of
communicating the output of the dynamic pricing policy has to be adapted to
the audience, be it humans with different levels of electricity market
expertise or algorithms (energy management systems).

###  3.7 Objective criterion

The dynamic pricing approach can provide multiple benefits, in terms of
economy, ecology but also autonomy. Consequently, the objective criterion to
be maximised by a dynamic pricing policy $\pi$ is not trivially determined. In
fact, several core objectives can be clearly identified:

  * $\bullet$

maximising the match between supply and demand,

  * $\bullet$

minimising the carbon footprint of power generation,

  * $\bullet$

minimising the electricity costs for the consumer,

  * $\bullet$

maximising the revenue of the producer/retailer.

Although some objectives overlap, these four criteria are not completely
compatible. For instance, maximising the synchronisation between power supply
and demand is equivalent to minimising the carbon footprint associated with
the generation of electricity. Indeed, the production portfolio of the
producer/retailer being mainly composed of intermittent renewable energy
sources, its energy has a reduced carbon footprint compared to the electricity
that can be purchased on the day-ahead market whose origin is unknown. On the
contrary, maximising the revenue of the producer/retailer will obviously not
lead to a minimised electricity bill for the consumer. This research work
makes the choice to prioritise the maximisation of the synchronisation between
supply and demand, and equivalently the minimisation of the carbon footprint,
while translating the other two core objectives into relevant constraints.
Firstly, the costs for the consumer have to be reduced with respect to the
situation without dynamic pricing. Secondly, the profitability of the
producer/retailer has to be guaranteed.

Formally, the objective criterion to be optimised by a dynamic pricing policy
$\pi$ can be mathematically defined as the following. First of all, the main
target to evaluate is the synchronisation between supply and demand, which can
be quantitatively assessed through the deviation $\Delta_{T}$. This quantity
has to ideally be minimised, and can be mathematically expressed as follows:

$\Delta_{T}=\sum_{t=0}^{T-1}|p_{t}-c_{t}|\text{,}$ (4)

where:

  * $\bullet$

$t=0$ corresponds to the first electricity delivery hour of a new day (00:00
AM),

  * $\bullet$

$T$ is the time horizon considered, which should be a multiple of 24 to have
full days,

  * $\bullet$

$p_{t}$ is the actual power production (not predicted) from the supply side at
time step $t$,

  * $\bullet$

$c_{t}$ is the actual power consumption (not predicted) from the demand side
at time step $t$.

Afterwards, the first constraint concerning the reduced costs for the consumer
has to be modelled mathematically. This is achieved via the electricity bill
$B_{T}$ paid by the consumer over the time horizon $T$, which can be expressed
as the following:

$B_{T}=\sum_{t=0}^{T-1}c_{t}\ y_{t}\ \text{.}$ (5)

As previously explained, the consumer power bill $B_{T}$ should not exceed
that obtained without dynamic pricing. In that case, the consumer is assumed
to pay a price $\overline{e_{t}}$, which can for instance be a fixed tariff or
a price indexed on the day-ahead market price. The situation without dynamic
pricing is discussed in more details in Section 5. Consequently, the first
constraint can be mathematically expressed as follows:

$\sum_{t=0}^{T-1}c_{t}\ y_{t}\leq\sum_{t=0}^{T-1}\overline{c_{t}}\
\overline{e_{t}}\ \text{,}$ (6)

where $\overline{c_{t}}$ is the power consumption from the demand side at time
step $t$ without dynamic pricing.

Then, the second constraint is about the profitability of the
producer/retailer, which is achieved if its revenue exceeds its costs. The
revenue $R_{T}$ of the producer/retailer over the time horizon $T$ can be
mathematically expressed as the following:

$R_{T}=\sum_{t=0}^{T-1}\left[c_{t}\ y_{t}-(c_{t}^{\prime}-p_{t}^{F})\
\lambda_{t}-(c_{t}-p_{t})\ i_{t}\right]\text{,}$ (7)

where:

  * $\bullet$

$\lambda_{t}$ is the actual power price (not predicted) on the day-ahead
market at time step $t$,

  * $\bullet$

$i_{t}$ is the actual imbalance price (not predicted) on the imbalance market
at time step $t$,

  * $\bullet$

$c_{t}^{\prime}$ is the predicted power consumption at time step $t$ after
demand response to the dynamic prices, based on the demand response
mathematical model $\mathcal{M}$.

The first term corresponds to the payment of the customers for their
electricity consumption. The second term is the revenue or cost induced by the
predicted mismatch between supply and demand, which is traded on the day-ahead
market. The last term is the cost or revenue caused by the remaining imbalance
between supply and demand, which has to be compensated in the imbalance
market.

The total costs incurred by the producer/retailer at each time step $t$ can be
decomposed into both fixed costs $F_{C}$ and marginal costs $M_{C}$. In this
particular case, the marginal costs of production are assumed to be negligible
since the production portfolio is composed of intermittent renewable energy
sources such as wind turbines and photovoltaic panels. Therefore, the second
constraint can be mathematically expressed as follows:

$\sum_{t=0}^{T-1}\left[c_{t}\ y_{t}-(c_{t}^{\prime}-p_{t}^{F})\
\lambda_{t}-(c_{t}-p_{t})\ i_{t}\right]\geq F_{C}\ T\ \text{.}$ (8)

Finally, the complete objective criterion to be optimised by a dynamic pricing
policy can be mathematically expressed as follows:

$\displaystyle\underset{\pi}{\text{minimise}}$
$\displaystyle\sum_{t=0}^{T-1}|p_{t}-c_{t}|\text{,}$ (9) subject to
$\displaystyle R_{T}\geq F_{C}\ T\ \text{,}$ $\displaystyle
B_{T}\leq\sum_{t=0}^{T-1}\overline{c_{t}}\ \overline{e_{t}}\ \text{.}$

##  4 Algorithmic components discussion

This section presents a thorough discussion about the different algorithmic
modules required to efficiently design a dynamic pricing policy from the
perspective of the supply side. Firstly, the different forecasting blocks are
rigorously analysed. Secondly, the modelling of the demand response induced by
dynamic prices is discussed. Lastly, the proper management of uncertainty is
considered.

In parallel, for the sake of clarity, Figure 3 highlights the interconnections
between the different algorithmic components in the scope of a dynamic pricing
policy from the perspective of the supply side. Moreover, Algorithm 1 provides
a thorough description of the complete decision-making process for the dynamic
pricing problem at hand. The complexity of the variable time lag between
decision-making and application is highlighted. Assuming that the decision-
making occurs once a day at 12:00 AM just before the closing of the day-ahead
market for all hours of the following day, the dynamic price at time step $t$
is decided hours in advance at time step $t-\left[12+(t\%24)\right]$ with the
symbol $\%$ representing the modulo operation.

Figure 3: Illustration of the complete decision-making process related to
dynamic pricing from the perspective of the supply side, with the connections
between the different algorithmic components highlighted. Algorithm 1 Dynamic
pricing complete decision-making process

The decision-making occurs once per day before the closing of the day-ahead
market at 12:00 AM for all hours of the next day.

The decision-making for the dynamic price of time step $t$ occurs at time step
$t-\left[12+(t\%24)\right]$.

for $\tau=-12$ to $T-12$ do

Check whether the time is 12:00 AM to proceed to the decision-making.

if $(\tau+12)\%24=0$ then

for $t=\tau+12$ to $\tau+36$ do

Gather the available information for production forecasting
$x_{t}^{P}=\\{W_{t}^{F},\ A_{t}^{F},\ I_{t}^{P}\\}$.

Gather the available information for consumption forecasting
$x_{t}^{C}=\\{W_{t}^{F},\ T_{t},\ I_{t}^{C}\\}$.

Gather the available information for day-ahead market price forecasting
$x_{t}^{M}=\\{x_{t}^{P},\ x_{t}^{C},\ G_{t}^{F},\ M_{t},\ I_{t}^{M}\\}$.

Forecast production at time step $t$:
$p_{t}^{F}=\mathcal{F}_{P}\left(x_{t}^{P}\right)$.

Forecast consumption at time step $t$:
$c_{t}^{F}=\mathcal{F}_{C}\left(x_{t}^{C}\right)$.

Forecast the day-ahead market price at time step $t$:
$\lambda_{t}^{F}=\mathcal{F}_{M}\left(x_{t}^{M}\right)$.

end for

for $t=\tau+12$ to $\tau+36$ do

Gather the input information for the dynamic pricing policy
$x_{t}=\\{P_{t}^{F},\ C_{t}^{F},\ \Lambda_{t}^{F},\ Y_{t},\ \mathcal{M}\\}$.

Make a dynamic pricing decision for time step $t$: $y_{t}=\pi(x_{t})$.

end for

Announce the dynamic prices for all hours of the following day $\\{y_{t}\ |\
t=\tau+12,...,\tau+35\\}$.

end if

end for

###  4.1 Production forecasting

The first forecasting block to be discussed concerns the production of
intermittent renewable energy sources such as wind turbines and photovoltaic
panels. Indeed, having access to accurate predictions about the future output
of the production portfolio is key to the performance of a dynamic pricing
policy from the perspective of the supply side. As previously explained in
Section 3.4, the forecasts have to be available one day ahead before the
closing of the day-ahead electricity market for all hours of the following
day. Naturally, the generation of such predictions introduces uncertainty, a
complexity that has to be taken into account to design sound dynamic pricing
policies.

Formally, the forecasting model associated with the output of the production
portfolio is denoted $\mathcal{F}_{P}$. Its input space $\mathcal{X}_{P}$
comprises every piece of information that may potentially have an impact on
the generation of electricity from intermittent renewable energy sources such
as wind turbines and photovoltaic panels for a certain time period. Its output
space $\mathcal{Y}_{P}$ is composed of a forecast regarding the power
generation from the production portfolio for that same time period.
Mathematically, the forecasting model input $x_{t}^{P}\in\mathcal{X}_{P}$ and
output $y_{t}^{P}\in\mathcal{Y}_{P}$ at time step $t$ can be expressed as
follows:

$x_{t}^{P}=\\{W_{t}^{F},\ A_{t}^{F},\ I_{t}^{P}\\}\text{,}$ (10)
$y_{t}^{P}=p_{t}^{F}\text{,}$ (11)

where:

  * $\bullet$

$W_{t}^{F}$ represents various weather forecasts related to the power
production of intermittent renewable energy sources such as wind turbines and
photovoltaic panels (wind speed/direction, solar irradiance, etc.) at the time
step $t$,

  * $\bullet$

$A_{t}^{F}$ represents predictions about the available capacity of the
production portfolio at time step $t$, which may be impacted by scheduled
maintenance, repairs, or other similar constraints,

  * $\bullet$

$I_{t}^{P}$ represents any additional information that may help to accurately
forecast the future power generation of the producer/retailer’s production
portfolio at time step $t$.

In the scientific literature, the current state of the art for forecasting the
power production of intermittent renewable energy sources is mainly based on
deep learning techniques together with some data cleansing processes and data
augmentation approaches. The best architectures are recurrent neural networks
(RNN), convolutional neural networks (CNN) and transformers [20, 21, 22, 23,
24].

###  4.2 Consumption forecasting

The objective of the next important forecasting model deserving a discussion
is to accurately predict the future power demand of the consumption portfolio
before any demand response phenomenon is induced. Since the main goal of a
dynamic pricing policy is to maximise the synchronisation between supply and
demand, electricity load forecasts are of equal importance to electricity
generation predictions. Similarly to the latter, the portfolio consumption
forecasts are assumed to be generated one day ahead just before the closing of
the day-ahead market for all 24 hours of the following day. Additionally, the
uncertainty associated with these predictions has to be seriously taken into
account for the success of the dynamic pricing policy.

From a more formal perspective, the forecasting model responsible for
predicting the future electricity load of the consumption portfolio is denoted
$\mathcal{F}_{C}$. Its input space $\mathcal{X}_{C}$ includes all the
information that may have an influence on the residential electricity
consumption for a certain time period. Its output space $\mathcal{Y}_{C}$
comprises a forecast of the power used by the consumption portfolio for that
same time period. Mathematically, the consumption forecasting model input
$x_{t}^{C}\in\mathcal{X}_{C}$ and output $y_{t}^{C}\in\mathcal{Y}_{C}$ at time
step $t$ can be expressed as the following:

$x_{t}^{C}=\\{W_{t}^{F},\ T_{t},\ I_{t}^{C}\\}\text{,}$ (12)
$y_{t}^{C}=c_{t}^{F}\text{,}$ (13)

where:

  * $\bullet$

$W_{t}^{F}$ represents various weather forecasts related to the residential
electricity consumption (temperature, hygrometry, etc.) at the time step $t$,

  * $\bullet$

$T_{t}$ represents diverse characteristics related to the time step $t$ (hour,
weekend, holiday, season, etc.),

  * $\bullet$

$I_{t}^{C}$ represents supplementary information that could potentially have
an influence on the residential power consumption at time step $t$.

Similarly to renewable energy production forecasting, the state-of-the-art
approaches for predicting the residential electricity load in the short term
are mostly related to deep learning techniques with preprocessed augmented
data: RNN, CNN, and transformers [25, 26, 27, 28, 22].

###  4.3 Market price forecasting

The last forecasting block to be discussed concerns the future day-ahead
electricity market prices. Contrarily to the forecasting of power production
and consumption, these price predictions are not critical to the success of a
dynamic pricing policy from the perspective of the supply side. Still, having
access to quality forecasts for the future day-ahead market prices remains
important in order to satisfy the constraints related to the profitability of
the producer/retailer as well as the reduced electricity costs for the
consumer. Once again, the predictions are assumed to be made just before the
closing of the day-ahead market. Moreover, the uncertainty associated with
these forecasts has to be taken into consideration.

Formally, the forecasting model related to the future day-ahead electricity
market prices is denoted $\mathcal{F}_{M}$. Its input space $\mathcal{X}_{M}$
includes every single piece of information which may potentially explain the
future electricity price on the day-ahead market for a certain hour. Its
output space $\mathcal{Y}_{M}$ comprises a forecast of the day-ahead market
price for that same hour. Mathematically, both forecasting model input
$x_{t}^{M}\in\mathcal{X}_{M}$ and output $y_{t}^{M}\in\mathcal{Y}_{M}$ at time
step $t$ can be expressed as follows:

$x_{t}^{M}=\\{x_{t}^{P},\ x_{t}^{C},\ G_{t}^{F},\ M_{t},\
I_{t}^{M}\\}\text{,}$ (14) $y_{t}^{M}=\lambda_{t}^{F}\text{,}$ (15)

where:

  * $\bullet$

$G_{t}^{F}$ represents forecasts about the state of the power grid as a whole
(available production capacity, transmission lines, etc.) at the time step
$t$,

  * $\bullet$

$M_{t}$ represents diverse information in various markets related to energy
(power, carbon, oil, gas, coal, etc.) in neighbouring geographical areas at
time step $t$,

  * $\bullet$

$I_{t}^{M}$ represents any extra piece of information that may help to predict
the future electricity price on the day-ahead market at time step $t$.

Once again, the scientific literature reveals that the state-of-the-art
approaches for day-ahead power market price forecasting are mostly based on
innovative machine learning techniques [29, 30, 31, 32, 33].

###  4.4 Demand response modelling

Another essential algorithmic component is the mathematical modelling of the
residential demand response to dynamic prices. In order to make relevant
dynamic pricing decisions, an estimation of the impact of the electricity
price on the consumer’s behaviour is necessary. In fact, two important
characteristics have to be studied:

The residential power consumption elasticity. This quantity measures the
average percentage change of the residential power consumption in response to
a percentage change in the electricity price. In other words, the elasticity
captures the willingness of the consumer to adapt its behaviour when the price
of electricity either increases or decreases. This elasticity is critical to
the dynamic pricing approach, since it assesses the receptiveness of the
consumers to dynamic prices. In fact, the residential power consumption
elasticity can be considered as a quantitative indicator of the potential of
the dynamic pricing approach.

The electricity load temporal dependence. Time plays an important role in
power consumption. Firstly, the consumer’s behaviour is highly dependent on
the time of the day. The tendency to adapt this behaviour is also expected to
be time-dependent. Therefore, the residential power consumption elasticity has
to be a function of the time within a day, among other things. Secondly, a
higher electricity price does not simply reduce the demand as with other
commodities, but rather shifts part of the consumption earlier and/or later in
time. This phenomenon reflects a complex temporal dependence for power
consumption, which has to be accurately modelled in order to design a
performing dynamic pricing policy.

Formally, the mathematical model of the residential demand response is denoted
$\mathcal{M}$. Its input space $\mathcal{X}_{D}$ is composed of the predicted
power consumption before any demand response and the dynamic prices to be sent
to the consumers for several hours before and after the time period analysed,
together with information about that time period. Its output space
$\mathcal{Y}_{D}$ comprises the predicted power consumption after demand
response to dynamic prices for that same time period. Mathematically, both
demand response model input $x_{t}^{D}\in\mathcal{X}_{D}$ and output
$y_{t}^{D}\in\mathcal{Y}_{D}$ at time step $t$ can be expressed as the
following:

$x_{t}^{D}=\\{C_{t}^{F},\ Y_{t}^{\prime},\ T_{t}\\}\text{,}$ (16)
$y_{t}^{D}=c_{t}^{\prime}\text{,}$ (17)

where $Y_{t}^{\prime}=\\{y_{t+\epsilon}\in\mathbb{R}\ |\ \epsilon=-k,...,k\\}$
is the dynamic price signal within a time window centred around time step $t$
and of size $k$ from which the demand response is induced.

As far as the scientific literature about the modelling of demand response to
dynamic prices is concerned, this interesting topic has not yet received a lot
of attention from the research community. Still, there exists a few sound
works presenting demand response models and assessing the receptiveness of the
consumers to dynamic power prices [15, 16, 17, 18, 19], as explained in
Section 2.

###  4.5 Uncertainty discussion

As previously hinted, a dynamic pricing policy has to make its decisions based
on imperfect information. Indeed, multiple forecasts for the electricity
price, production and consumption have to be generated 12 up to 35 hours in
advance. Naturally, these predictions comes with a level of uncertainty that
should not be neglected. Moreover, accurately modelling the residential demand
response to dynamic prices is a particularly challenging task. Because of both
the random human nature and the difficulty to fully capture the consumers’
behaviour within a mathematical model, a notable level of uncertainty should
also be considered at this stage. Therefore, multiple sources of uncertainty
can be identified in the scope of the dynamic pricing decision-making problem
at hand, and a proper management of this uncertainty is necessary.

A stochastic reasoning is recommended to make sound dynamic pricing decisions
despite this substantial level of uncertainty. Instead of considering each
uncertain variable (production, consumption, price, demand response) with a
probability of 1, the full probability distribution behind these quantities
has to be estimated and exploited. Based on this information, the risk
associated with uncertainty may be mitigated. Moreover, safety margins may
also contribute to reduce this risk, but potentially at the expense of a
lowered performance. In fact, there generally exists a trade-off between
performance and risk, in line with the adage: with great risk comes great
reward.

##  5 Performance assessment methodology

This section presents a methodology for quantitatively assessing the
performance of a dynamic pricing policy in a comprehensive manner. As
explained in Section 3.7, several disjoint objectives can be clearly
identified. For the sake of completeness, this research work presents three
quantitative indicators, one for each objective. The relative importance of
these indicators is left to the discretion of the reader according to its main
intention among the different objectives previously defined.

The performance indicators proposed are based on the comparison with the
original situation without dynamic pricing. In this case, the consumer is
assumed to be fully ignorant about the mismatch problem between supply and
demand. No information is provided to the customers of the producer/retailer,
which consequently have an uninfluenced consumption behaviour. The price of
electricity $\overline{e_{t}}$ is freely determined by the producer/retailer.
It may for instance be a fixed tariff, or a price indexed on the day-ahead
market price:

$\overline{e_{t}}=\alpha\ \lambda_{t}+\beta\ \text{,}$ (18)

where $\alpha$ and $\beta$ are parameters to be set by the retailer.

Firstly, the impact of a dynamic pricing policy on the synchronisation between
power supply and demand can be assessed through the performance indicator $S$
quantifying the relative evolution of the deviation $\Delta_{T}$. This
quantity is mathematically expressed as follows:

$S=100\ \frac{\overline{\Delta_{T}}-\Delta_{T}}{\overline{\Delta_{T}}}\
\text{,}$ (19)
$\overline{\Delta_{T}}=\sum_{t=0}^{T-1}|p_{t}-\overline{c_{t}}|\ \text{,}$
(20)

where $\overline{\Delta_{T}}$ represents the lack of synchronisation between
supply and demand without dynamic pricing. Therefore, the quantity $S$ has
ideally to be maximised, with a perfect synchronisation between supply and
demand leading to a value of $100\%$ reduction in deviation.

Secondly, the consequence for the consumer regarding its electricity bill can
be evaluated with the quantity $B$ which informs about the relative evolution
of this power bill. It can be mathematically computed as the following:

$B=100\ \frac{\overline{B_{T}}-B_{T}}{\overline{B_{T}}}\ \text{,}$ (21)

where $\overline{B_{T}}=\sum_{t=0}^{T-1}\overline{c_{t}}\ \overline{e_{t}}$
represents the electricity bill paid by the consumer without dynamic pricing.
Since the performance indicator $B$ represents the percentage reduction in
costs, it has to ideally be maximised.

Lastly, the enhancement in terms of revenue for the producer/retailer can be
efficiently quantified thanks to the performance indicator $R$. This quantity
represents the relative evolution of the producer/retailer revenue and can be
mathematically expressed as follows:

$R=100\ \frac{R_{T}-\overline{R_{T}}}{\overline{R_{T}}}\ \text{,}$ (22)
$\overline{R_{T}}=\sum_{t=0}^{T-1}\left[\overline{c_{t}}\
\overline{e_{t}}-(c_{t}^{F}-p_{t}^{F})\ \lambda_{t}-(\overline{c_{t}}-p_{t})\
i_{t}\right]\text{,}$ (23)

where $\overline{R_{T}}$ represents the producer/retailer revenue without
dynamic pricing. Obviously, the performance indicator $R$ has to ideally be
maximised.

##  6 Conclusion

This research paper presents a detailed formalisation of the decision-making
problem faced by a producer/retailer willing to adopt a dynamic pricing
approach, in order to induce an appropriate residential demand response. Three
core challenges are highlighted by this formalisation work. Firstly, the
objective criterion maximised by a dynamic pricing policy is not trivially
defined, since different goals that are not compatible can be clearly
identified. Secondly, several complex algorithmic components are necessary for
the development of a performing dynamic pricing policy. One can for instance
mention different forecasting blocks, but also a mathematical model of the
residential demand response to dynamic prices. Thirdly, the dynamic pricing
decisions have to be made based on imperfect information, because this
particular decision-making problem is highly conditioned by the actual
uncertainty for the future.

Several avenues are proposed for future work. In fact, the natural extension
of the present research is to design innovative dynamic pricing policies from
the perspective of the supply side based on the formalisation performed. While
the present research paper exclusively focuses on the philosophy and
conceptual analysis of the approach, there remain practical concerns that need
to be properly addressed in order to achieve performing decision-making
policies. To achieve that, a deeper analysis of the scientific literature
about each algorithmic component discussed in Section 4 is firstly welcomed,
in order to identify and reproduce the state-of-the-art techniques within the
context of interest. Then, different approaches have to be investigated for
the design of the dynamic pricing policy itself. One can for instance mention,
among others, the stochastic optimisation and deep reinforcement learning
techniques. Finally, the dynamic pricing policies developed have to be
rigorously evaluated, analysed, and compared by taking advantage of real-life
experiments.

## Acknowledgements

Thibaut Théate is a Research Fellow of the F.R.S.-FNRS, of which he
acknowledges the financial support.

## References

  * [1] IPCC, Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 2021.
  * [2] H. Ritchie, M. Roser, Energy, Our World in Data, https://ourworldindata.org/energy (2020).
  * [3] S. Chatzivasileiadis, D. Ernst, G. Andersson, The global grid, Renewable Energy 57 (2013) 372–383.
  * [4] N. Kittner, F. Lill, D. M. Kammen, Energy storage deployment and innovation for the clean energy transition, Nature Energy 2 (2017) 17125.
  * [5] P. Palensky, D. Dietrich, Demand side management: Demand response, intelligent energy systems, and smart loads, IEEE Transactions on Industrial Informatics 7 (2011) 381–388.
  * [6] P. Siano, Demand response and smart grids — A survey, Renewable & Sustainable Energy Reviews 30 (2014) 461–478.
  * [7] R. Deng, Z. Yang, M.-Y. Chow, J. Chen, A survey on demand response in smart grids: Mathematical models and approaches, IEEE Transactions on Industrial Informatics 11 (2015) 570–582.
  * [8] J. S. Vardakas, N. Zorba, C. V. Verikoukis, A survey on demand response programs in smart grids: Pricing methods and optimization algorithms, IEEE Communications Surveys & Tutorials 17 (2015) 152–178.
  * [9] H. T. Haider, O. H. See, W. Elmenreich, A review of residential demand response of smart grid, Renewable & Sustainable Energy Reviews 59 (2016) 166–178.
  * [10] Z. Zhao, W. C. Lee, Y. Shin, K.-B. Song, An optimal power scheduling method for demand response in home energy management system, IEEE Transactions on Smart Grid 4 (2013) 1391–1400.
  * [11] M. Muratori, G. Rizzoni, Residential demand response: Dynamic energy management and time-varying electricity pricing, IEEE Transactions on Power Systems 31 (2016) 1108–1117.
  * [12] N. Liu, X. Yu, C. Wang, C. Li, L. Ma, J. Lei, Energy-sharing model with price-based demand response for microgrids of peer-to-peer prosumers, IEEE Transactions on Power Systems 32 (2017) 3569–3583.
  * [13] J. R. Vázquez-Canteli, Z. Nagy, Reinforcement learning for demand response: A review of algorithms and modeling techniques, Applied Energy (2019).
  * [14] H. Li, Z. Wan, H. He, Real-time residential demand response, IEEE Transactions on Smart Grid 11 (2020) 4144–4154.
  * [15] S. Gottwalt, W. Ketter, C. Block, J. Collins, C. Weinhardt, Demand side management — A simulation of household behavior under variable prices, Energy Policy 39 (2011) 8163–8174.
  * [16] M. Brégère, R. J. Bessa, Simulating tariff impact in electrical energy consumption profiles with conditional variational autoencoders, IEEE Access 8 (2020) 131949–131966.
  * [17] K. Ganesan, J. T. Saraiva, R. J. Bessa, Functional model of residential consumption elasticity under dynamic tariffs, Energy and Buildings 255 (2022).
  * [18] Y. He, B. Wang, J. Wang, J. Wang, W. Xiong, T. Xia, Residential demand response behavior analysis based on Monte Carlo simulation: The case of Yinchuan in China, Energy 47 (2012) 230–236.
  * [19] E. A. M. Klaassen, C. B. A. Kobus, J. Frunt, J. G. Slootweg, Responsiveness of residential electricity demand to dynamic tariffs: Experiences from a large field test in the Netherlands, Applied Energy 183 (2016) 1065–1074.
  * [20] C. Sweeney, R. J. Bessa, J. Browell, P. Pinson, The future of forecasting for renewable energy, WIREs Energy and Environment (2019).
  * [21] R. Ahmed, V. Sreeram, Y. D. Mishra, M. Arif, A review and evaluation of the state-of-the-art in PV solar power forecasting: Techniques and optimization, Renewable & Sustainable Energy Reviews 124 (2020) 109792.
  * [22] S. Aslam, H. Herodotou, S. M. Mohsin, N. Javaid, N. Ashraf, S. Aslam, A survey on deep learning methods for power load and renewable energy forecasting in smart microgrids, Renewable and Sustainable Energy Reviews 144 (2021).
  * [23] H. Jahangir, H. Tayarani, S. S. Gougheri, M. A. Golkar, A. Ahmadian, A. Elkamel, Deep learning-based forecasting approach in smart grids with microclustering and bidirectional LSTM network, IEEE Transactions on Industrial Electronics 68 (2021) 8298–8309.
  * [24] D. Heinemann, E. Lorenz, M. Girodo, Forecasting solar radiation, Journal of Cases on Information Technology (2021).
  * [25] W. Kong, Z. Y. Dong, Y. Jia, D. J. Hill, Y. Xu, Y. Zhang, Short-term residential load forecasting based on LSTM recurrent neural network, IEEE Transactions on Smart Grid 10 (2019) 841–851.
  * [26] N. Somu, G. R. M. R, K. Ramamritham, A hybrid model for building energy consumption forecasting using long short term memory networks, Applied Energy 261 (2020) 114131.
  * [27] X. bo Jin, W. Zheng, J. Kong, X. Wang, Y. Bai, T. Su, S. Lin, Deep-learning forecasting method for electric power load via attention-based encoder-decoder with bayesian optimization, Energies 14 (2021) 1596.
  * [28] A. Gasparin, S. Lukovic, C. Alippi, Deep learning for time series forecasting: The electric load case, CAAI Transactions on Intelligence Technology 7 (2021).
  * [29] R. Weron, Electricity price forecasting: A review of the state-of-the-art with a look into the future, HSC Research Reports (2014).
  * [30] J. Nowotarski, R. Weron, Recent advances in electricity price forecasting: A review of probabilistic forecasting, HSC Research Reports (2016).
  * [31] A. R. Gollou, N. Ghadimi, A new feature selection and hybrid forecast engine for day-ahead price forecasting of electricity markets, J. Intell. Fuzzy Syst. 32 (2017) 4031–4045.
  * [32] U. Ugurlu, I. Oksuz, O. Tas, Electricity price forecasting using recurrent neural networks, Energies 11 (2018) 1255.
  * [33] H. Jahangir, H. Tayarani, S. Baghali, A. Ahmadian, A. Elkamel, M. A. Golkar, M. Castilla, A novel electricity price forecasting approach based on dimension reduction strategy and rough artificial neural networks, IEEE Transactions on Industrial Informatics 16 (2020) 2369–2381.