Energy Management of a Battery Storage System Considering Variable
Load and Controllable Renewable Generation (Solar Study Case) to
Keep the Grid’s Frequency Stability
L. C. PEREZ1, L. A. GARCIA2, J. HERNANDEZ-COBA3, S. R. RIVERA4
1Facultad de Ingeniería Mecánica y Eléctrica,
Universidad Autónoma de Nuevo León,
MEXICO
2Departamento de Ingeniería Eléctrica,
Universidad Nacional de Colombia,
COLOMBIA
3IEE-UNSJ-CONICET
Universidad Nacional de San Juan
ARGENTINA
4Departamento de Ingeniería Eléctrica
Universidad Nacional de Colombia
COLOMBIA
Abstract: - This paper presents the research of analytical functions related to the energy generation of
photovoltaic systems and the residential and commercial load demanded by end users, concerning a statistical
function. To test this model, a linear cost function was considered to compute its overestimation and
underestimation due to its maximum and minimum production limits, where energy consumption is obtained at
each instant of time, within the established production ranges, through the analytical equations that determine
solar energy generation and demand load. The result obtained by applying the Uncertainty Quantification
(UCF) theory in these equations, in the same way through the Monte Carlo (MC) simulation for comparison, is
the expected value of energy for a hypothetical storage system E (Cu, Co). Better accuracy of results via this
model can be improved upon when the energy generation parameters are structured as analytical functions each
instant of time associated with probability distributions based on the uncertainty costs of controllable sources,
instead of statistical functions.
Key-Words: - energy management, frequency, mathematical modeling, Montecarlo simulation, overestimation
cost, solar photovoltaic, Uncertainty Cost Function, underestimation cost.
Received: March 15, 2023. Revised: December 8, 2023. Accepted: December 24, 2023. Published: December 31, 2023.
1 Introduction
The main objective of energy management is to
ensure the safe, reliable, and efficient operation of
power systems. Reducing peak demand where there
is greater consumption by end users is one of
several approaches that contribute to the
stabilization of a system. Another method is the
production of energy through renewable sources in
microgrids, as they provide a means to stabilize the
grid’s frequency, [1].
Microgrids are defined as a group of distributed
loads and generators that operate as a controllable
unit that provides power to its area either in
isolation or connected to the conventional power
network, [2], In paper [3], a comprehensive review
of the current status of microgrids is presented,
which discusses design trends, challenges and
research efforts towards their implementation in
power systems.
Energy generation in microgrids depends on the
stochastic behavior of the renewable sources that
comprise them, which might affect various
parameters, including frequency. To address the
issue of frequency deviation in microgrid control,
various control strategies and methods have been
proposed in recent research. For instance, the
authors in [4], introduce a fractional order
proportional-integral-derivative (FOPID) controller
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L. C. Perez, L. A. Garcia, J. Hernandez-Coba, S. R. Rivera
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for islanded microgrids using intelligent
optimization algorithms, which demonstrated
effective frequency control. Additionally, in [5], the
necessity of load shedding control schemes to
maintain power balance and frequency stability in
islanded microgrids is emphasized.
Frequency stability refers to the system’s ability
to maintain a constant nominal electrical frequency
within acceptable limits, even in the face of
disturbances or changes in load. To maintain system
stability regarding frequency, it is fundamental to
balance energy supply (generation) and demand in
real time, [6].
In this context, the authors in [7], present
Uncertainty Cost Functions (UCF) to model and
evaluate stochasticity in power systems with high
penetration of renewable energy sources, where the
functions of uncertainty costs have analytically
calculated minimum cost values and the marginal
derived cost functions (MUCF) can be used as
inputs for economic dispatch and optimal power
flow (OPF) calculations, which support frequency
stability.
Related research focusing on the OPF problem
formulation has considered restrictions dealing with
the secure operation of power systems in order to
keep a balance between generation and demand in
post-contingency states, [8], [9]. Specifically, the
methodology to solve a probabilistic Security-
Constrained Optimal Power Flow (SCOPF) to
assess N-k contingencies is detailed in [8]. In [9], an
iterative algorithm for solving realistic SCOPF
problems in large-scale power systems is presented
and its main features are discussed, i.e.,
consideration of nonlinear AC network models in
both pre-contingency and post-contingency states,
and optimization of active/reactive power flows
jointly.
The before mentioned contingency analyses have
an application in the framework of short-term asset
management, also known as real-time asset
management, [10], [11]. This is a detailed
assessment of the possible impacts an unexpected
outage might have on a certain asset’s condition and
performance. Its results support grid operators in the
decision-making process concerning the definition
of post-contingency states.
As the participation of renewable energy sources
(e.g., solar and wind) in the generation mix of power
systems has increased steadily in the last years,
different approaches have been proposed to update
the formulation of OPF problems. One such
proposal is the adaptive geometry estimation-based
multi-objective differential evolution (AGE-MODE)
method for multi-objective OPF in hybrid power
systems, which considers the stochastic behavior of
solar photovoltaic (PV) and wind through
probability distribution functions to compute direct
costs, penalty costs for underestimation, and reserve
costs for overestimation, [12].
Regarding the costs (i.e., under- and
overestimation costs) associated with the
intermittency and variability over time of generation
based on renewable energy sources, different papers
have analyzed the potential of UCF to enhance the
mathematical formulation of OPF problems. For
example, the authors in [13], propose the application
of UCF in the economic dispatch of power systems
with a penetration of small hydroelectric plants
(SHPs). To this end, the analytical development of
the UCF based on the power injected by a SHP is
presented, as well as the validation procedure of the
computed under- and overestimation costs via a
Monte Carlo (MC) simulation.
Similar research has developed UCF for the
power consumed by electric vehicles and the power
output of solar PV and wind plants, [14], [15]. The
detailed mathematical approach to compute the
under- and overestimation costs was also validated
using a MC simulation.
As it has been explained, different studies have
analyzed the enhancement of OPF problems and,
particularly, the ability of UCF to model the
stochastic behavior inherent to the power output of
renewable energy sources as a means to improve the
formulation of the before mentioned problems.
However, for the authors’ best knowledge, any of
these studies have analyzed the potential of UCF to
support the frequency control of power systems in a
context of increased solar PV generation.
For this reason, this article introduces the use of
UCF to estimate an expected energy value that
should be stored or released in a Battery Storage
System (BSS) to control the stability of the grid’s
frequency. This methodology has two parts, i.e., an
analytical development to calculate the expected
energy value and a MC simulation. The obtained
quantities are compared and the percentage error is
computed to validate the accuracy of the analytical
proposed UCF.
The structure of the remaining part of the paper
is as follows: section 2 explains the selected
methodology; section 3 covers the data analysis,
study case, and application of results; and section 4
deals with the discussion of results and concluding
remarks.
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2 Problem Formulation
Renewable energy sources like wind and solar
exhibit variable behaviors due to their dependence
on weather conditions. This variability can lead to
fluctuations in electrical grid frequency, causing
instability and equipment damage. Frequency
variation, ∆f(t), is linked to the power system’s
demand and generation, D(t) and G(t), respectively,
as indicated by the following equation:
(1)
where K is a constant that represents the power
system’s inertia. If an ideal power system is
considered, i.e., losses in conductors are not
disregarded, the total demand must match the output
of all available generators at each time instant. This
ensures a stable frequency, i.e., ∆f(t) = 0, and
Equation 2 is obtained if it is assumed the
generation of electrical energy is provided just by
solar PV, Gsun(t), and wind plants, Gwind(t).
(2)
As it is mentioned previously, renewable energy
sources have a stochastic or variable output and a
Battery Storage System (BSS) is required to ensure
frequency stability. The amount of electrical energy
(Eb) that should be stored or released from the BSS
within a time range (t1, t2) is computed by solving
Equation 3.
(3)
The proposed methodology requires an
analytical function with sinusoidal behavior that
represents the energy output of a PV-system as a
function of time, referenced to the time of sunrise
(trise) and sunset (tset) during the day. Another
function represents the power demanded by a
residential and industrial load, considering the
before mentioned time variables and a change
concerning the peak demand during the morning
and afternoon. The energy in MWh for both
functions could be determined calculating the area
below the curve, i.e., calculating the integral.
2.1 Power Supplied by a PV-Plant as a
Function of Time
The daily power output of a PV-plant as a function
of time can be expressed analytically as:
(4)
where the variation in the intensity of sunlight is
modeled during the period between trise and tset. The
equation follows the radiation curve, which is
modeled as a sinusoidal function, having some
variation during the day. This variation for practical
purposes and for the development of the paper, a
Pmax and Pmin will be analyzed, reaching their
maximum powers around noon and being minimum
at dawn and dusk, respectively, in their established
ranges.
(5)
(6)
These functions are normalized so that the
power of each function varies from 0 to a maximum
within allowable ranges. The power generated by a
solar panel or solar system depends on many
factors, such as geographical location, panel
inclination, weather conditions, among others.
However, for this analysis it was simplified by
providing random values.
2.2 Power Required by the Selected Load as
a Function of Time
The power demanded in a residential or industrial
load can vary throughout the day, and its load
profile typically follows predictable patterns. The
analytical expression of the power demanded as a
function of time is represented as:
(7)
This function considers a daily variation with
different patterns during the day, where the equation
models the electrical demand with a constant base
component and two sinusoidal components that
represent the demand peaks in the morning and
afternoon. It is necessary to adjust the parameters of
the equation depending on the specific consumption
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patterns of the residential or industrial load being
modeled.
2.3 Uncertainty Quantification
The mathematical development obtained using the
uncertainty quantification theory, considering in
each time instant D and Gsun, which have associated
some probability distributions, would provide an
expected value of E.
(8)
(9)
The previous results make it possible to
calculate the expected uncertainty cost function
(UCF), which describes a remarkable quadratic
pattern, something useful for conventional economic
dispatch software.
(10)
The uncertainty cost can be modeled as a
function of time performing simple calculations.
3 Study Case and Simulations
3.1 Monte Carlo simulation
Monte Carlo (MC) simulation uses random
sampling and statistical modeling to estimate
mathematical functions and mimic the operations of
complex systems. When applied in physical models,
this method generates data from fixed probability
distribution functions of stochastic variables such as
solar irradiance, customer demands, etc., [16], [17],
and it has gained widespread acceptance to validate
their accuracy, [15], [18].
For this reason, this research considers MC
simulation to study the behavior of overestimation
and underestimation instances for a predetermined
power value (Ps), within a set of power values
uniformly distributed over a 24-hour range. The test
values initially set for analysis were the variables Ps
(average power), Pmax (maximum power), and Pmin
(minimum power) at 100, 110, and 90 watts,
respectively. The uncertainty costs of
underestimation and overestimation were adopted
from reference [4], with Cu=300 and Co=700 values,
respectively.
Equations 4, 5, and 6 represent the demanded
powers of a residential or industrial load, wherein
the sunrise time (trise) was considered at 6 AM,
while the sunset time (tset) was taken as 6 PM. These
values were considered for practical purposes;
however, they can be analytically obtained by
considering various factors such as incidence angle,
extraterrestrial radiation, climate type, solar
declination, among others. Figure 1 illustrates the
power values derived from Equations 4, 5, and 6
concerning the predetermined time.
Fig. 1: Solar power
Equation 11 will generate random values within
the interval (Pmin, Pmax) following a uniform
distribution.
(11)
The outcome will establish values within the
generated scenarios (N=1000) for the
Overestimation Cost (Co) and Underestimation Cost
(Cu), depending on the average power value (Ps), as
depicted in Figure 2.
(12)
(13)
Following an elapsed simulation time of
approximately 0.18 seconds, multiple statistical
parameters were obtained.
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a)
b)
Fig. 2: Behavioral curve of solar generation
throughout the day, including: a) random scenario
of solar generation and b) Monte Carlo simulation
curves in scenarios with maximum and minimum
values.
The histogram in Figure 3 represents the sum of
all generated powers across the N scenarios,
exhibiting a high frequency around 600 MW. The
estimated energy output resulted in 599.82 MWh.
Fig. 3: Expected energy from MC scenarios
3.2 Uncertainty Cost Functions
MC simulation was employed to derive
underestimation and overestimation penalty values
for the Uncertainty Cost (UC) of photovoltaic
generation at a specific time instance. Under a
uniform distribution model, the validation for the
Uncertainty Cost Function (UCF) was presented and
compared favorably with MC simulation,
showcasing minimal error.
where,
(14)
The average of the sum of
and the result from the
Analytical Hourly Uncertainty Cost Function
(UCFPAH) in Equation 14 is illustrated in Table 1,
demonstrating minimal error percentage.
Table 1. Comparison of results between the
analytical method and the Montecarlo method
Time
(Hour)
UCF MEAN
(MW/$)
UCFPAH
(MW/$)
%error
7
0.1725
0.1674
0.2953
8
0.6165
0.6250
0.1367
9
1.2657
1.2499
0.1246
10
1.8341
1.8749
0.2228
11
2.4129
2.3325
0.3331
12
2.5283
2.5000
0.1121
13
2.4040
2.3325
0.2974
14
1.8958
1.8750
0.1102
15
1.2838
1.2500
0.2640
16
0.6149
0.6250
0.1632
17
0.1673
0.1674
0.0045
Given text already adheres to the principles.
Considering that it is photovoltaic solar generation,
the other hours of the series have a value of zero.
On the other hand, the variable k is associated
with the uncertainty cost function, either hourly
estimated costs or analytical hourly costs, within the
24-hour time range.
(15)
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The average KPAHprom, shown in Figure 4, serves
as a link for MWh energy estimation in an analytical
model, affirming its similarity. It is noteworthy that
only non-zero values are taken into consideration.
(16)
Fig. 4: Constant KPAHprom
Figure 5 shows the values from Table 1,
comparing the analytical and Monte Carlo methods
graphically, with a minimum error observed.
Fig. 5: Comparison of UCF, analytical and Monte
Carlo simulation
3.3 Analytical Method
An analytical method represents a systematic
approach employed to comprehend, explain, or
solve problems by utilizing analysis, logical
reasoning, and, often, mathematical formulas or
existing theories. It is characterized by its emphasis
on breaking down a problem into smaller, more
manageable parts for detailed examination. These
methods enable the dissection of complex issues
into simpler components, facilitating their
comprehension and resolution.
The necessity of providing a model for load and
generation with variability in behavior becomes
evident when estimating costs. Therefore, an
analytical proposal will be developed to estimate the
energy in MWh that the photovoltaic system will
produce over 24 hours.
For a linear function representing the cost of
underestimation penalty, determining the
corresponding expected penalty cost function can be
achieved as follows:
where,
(17)
To obtain the energy produced over time, the
function is integrated to find the area under the
curve within the established range (6, 18), denoting
sunrise and sunset hours.
(18)
where,
(19)
Similarly, the expected cost function for
overestimation with E[Co(P)] can be derived:
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where,
(20)
(21)
where,
(22)
The results enable the calculation of the
expected UCF for both overestimation and
underestimation, whose outcome, multiplied by the
variable Kprom obtained from Monte Carlo
simulation, allows us to calculate the energy in
MWh. The result from the analytical function is
600 MWh, which is like the result obtained with the
Monte Carlo simulation 599.85 MWh.
4 Discussion of Results and Conclusion
The results of the previous sections, where the
Monte Carlo simulations and the analytical results
shown in Table 1 clearly show that the results are
equivalent with estimation errors of less than 0.34%,
demonstrating the possibility of using the analytical
method for energy estimation.
This result makes it possible to consider the
implementation of algorithms for calculating energy
to manage storage in batteries in scenarios where
there are variable loads.
On the contrary, regarding the estimation of
variable KPAHprom that relates to the cost function of
uncertainty, forthcoming research could explore the
use of a polynomial function approximation capable
of capturing changes in uncertainty magnitude in
difference time instances.
Among the conclusions, the following are
highlighted:
The analytical equations for estimating the
energy in loads and in solar photovoltaic
generation are equivalent to the log-normal
statistical function considered.
The finding of equivalence makes it
possible to apply this new method for
energy management in hybrid systems
where there is solar generation and batteries
for storage.
The initial hypothesis about the possibility
of maintaining the frequency from the
energy equivalence is fulfilled, leading to
improvements in the response of the
algorithms.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Conceptualization S.R.R.; methodology, , L.C.P,
L.A.G, J.H.C and S.R.R.; software, L.C.P and
S.R.R.; validation, L.C.P, L.A.G, J.H.C and S.R.R.;
formal analysis, , L.C.P, L.A.G, J.H.C and S.R.R.;
investigation, S.R.R.; resources, , L.C.P, L.A.G,
J.H.C and S.R.R.; data curation, L.C.P and S.R.R.;
writing—original draft preparation, S.R.R.;
writing—review and editing, L.C.P, L.A.G, and
J.H.C.; visualization, , L.C.P.; supervision, S.R.R.;
project administration, , S.R.R.. The authors have
read and agreed to the published version of the
manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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DOI: 10.37394/232016.2023.18.40
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