Quantifying Uncertainty Costs in Renewable Energy Systems
Considering Probability Function Behavior and CVaR at Low-
Probability Generation Extremes using Deterministic Equations
L. C. PEREZ1, G. IDARRAGA-OSPINA1, S. R. RIVERA2
1Facultad de Ingeniería Mecánica y Eléctrica
Universidad Autónoma de Nuevo León
MEXICO
2Departamento de Ingeniería Eléctrica
Universidad Nacional de Colombia
COLOMBIA
Abstract: - The increase and integration of renewable energy sources in electrical power systems implies an
increase in uncertainty variables, both in costs and production, of economic dispatch (ED) and currently have a
significant influence on wholesale electricity markets (MEM). Uncertainty costs refer to the quantification of
additional expenses or economic losses associated with the variability inherent in the generation of renewable
energy, such as wind, solar, or hydroelectric. Therefore, this article presents deterministic equations related to
cost overestimation and underestimation, as well as CVaR, to model and evaluate the stochasticity of risks
associated with the integration of renewable sources, allowing system operators and planners to make informed
decisions. To mitigate or use said risks in energy systems with high penetration of elements, mainly smart
networks. In this study, a mathematical analysis is carried out using the histogram spectrum formed by the
power generated by the probability density function (PDF) for solar generation, although it is possible to
consider other types of functions to determine energy generation. The objective of the proposed model is to
provide another tool to the system operator for energy management and planning, which relieves a little of the
weight of the computational load and at the same time presents more precision in the results by being able to
work with a database. Historical data if these values are available. Commonly, for this type of analysis, values
are estimated using probabilistic calculations by density functions when integrating these functions, or in other
recent cases by estimating them by analytical methods of the same functions. A validation of the model is
presented by comparing the result with the Monte Carlo simulation, developing the total cost of uncertainty
only from "low probability generation extremes". Furthermore, the results are presented through analytical
uncertainty cost functions (AUCF). This analysis includes the calculation of uncertainty costs for low and high-
probability energy generation, determined by the Conditional Value at Risk (CVaR), using deterministic
equations.
Key-Words: - analytical uncertainty, conditional value at risk, economic dispatch, histogram, low probability,
mathematical modeling, Monte Carlo, probability density function, uncertainty cost, risk.
Received: April 14, 2024. Revised: September 7, 2024. Accepted: October 11, 2024. Published: November 13, 2024.
1 Introduction
The integration of renewable energy sources into
power systems is a crucial aspect of transitioning
towards a more sustainable and resilient energy
matrix. Renewable energy generation, such as solar
and wind, possesses unique characteristics that can
impact the stability and operation of electrical
systems. These renewable energy sources (RESs)
are becoming increasingly important to mitigate the
environmentally harmful impacts of traditional
energy sources. Despite the inherent advantages of
pollution reduction and resource conservation, the
inherent variability in renewable generation,
influenced by factors like natural resource
availability and weather conditions, poses
challenges in the operation, and distribution of the
power system and in terms of managing energy
supply and demand, [1].
The main objective of using renewable sources
in the main power system, in addition to reducing
gas emissions and replacing conventional
generation, is cost reduction. However, the use of a
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L. C. Perez, G. Idarraga-Ospina, S. R. Rivera
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single renewable technology is not sufficient for the
large daily demand, the same occurs if only a single
main source of energy is used, due to its variability
and imprecision in its generation.
The authors in [2] have suggested integrated
scheduling of wind, thermal, and hydropower
including spinning reserve. Several literature opt for
sets of renewable sources, which is why the
continuity of the electrical service is more efficient
than opting only for solar and wind sources due to
their uncertainty. In [3], the minimization of
the power generation cost of the thermal power units
is achieved by incorporating renewable sources,
such as hydro, winds, and solar plants for 24 hours
scheduled, and available transfer capability
calculation is the prime objective. This mixture of
non-conventional generations usually provides
greater support to the network but is not exempt
from improvements when replacing or adding other
sources. Today there is no completely reliable
method or technique for the integration of RESs into
the network, however, several kinds of the literature
raise the problem in the short, medium, and long
term in real-time.
In [4], [5], they use the same methodology,
however, they highlight the use of stochastic or
deterministic optimization techniques. The article
presents a short-term hydrothermal-wind
complementary scheduling system (HTWCS)
considering the uncertainty of wind energy, as well
as several non-linear systems, formulated as a multi-
objective optimization problem to optimize
economic and environmental criteria. An improved
multi-objective bee colony optimization
(EMOBCO) algorithm is proposed to solve this
problem, and the second paper a hydro-thermal-
wind-solar hybrid energy system of the provincial
power grid is taken as the background. A practical
method for long-term coordination for this system is
proposed.
In real-time economic dispatch (ED), due to
accurate forecasting, the range of uncertainty is
lower compared to the long-run scheduling, [6]. The
economic dispatch of energy on power systems with
high penetration of renewable generation is a
mathematical problem of optimization. The paper
[7], shows a mathematical analysis with
probabilistic methods contrasted with an analytic
development for controllable renewable systems to
be included in the target functions of economic
dispatch problems.
In [8], analytical formulas of uncertainty penalty
costs are calculated, for solar and wind energy and
electric vehicles, through a mathematical expected
value formulation.
The integration of renewable sources into the
grid opens the way to various methods for their
interconnection without affecting the stability of the
main system, as well as new concepts such as costs
due to the overestimation and underestimation
presented by the uncertainties of renewable sources.
The authors in [7], [8], [9] and [10], work on this
concept analytically, improving the time and
precision of the results that are commonly worked
on by mathematical methods such as the Monte
Carlo Method supported by the functions proposed
for each generation. They present an analysis in the
development of a new mathematical formulation
with which it will be possible to determine, through
probabilistic approaches, the cost that can be
generated if a diversified electricity market exists, in
which the demand can actively participate.
Uncertain behavior of renewable generation
plants and PEV modeling can be done by
probability distribution functions (PDFs), as shown
in [11], where the wind speed for the plants was
modeled by the Weibull PDF, and the solar
irradiance was modeled by a lognormal distribution.
Probability Density Functions (PDFs) for
determining solar and wind power are statistical
tools that describe the distribution of energy
generated by renewable sources based on the
probability of occurrence. These functions allow
modeling and predicting the variability in solar and
wind energy production, which is crucial for the
design and efficient management of renewable
energy systems.
This article mentions normal and log-normal
PDF for solar energy estimation and Weibull PDF
for wind energy estimation. The main objective is
the determination of the costs of uncertainties
through a deterministic method, using as a
calculation basis the histogram generated by the
powers that are determined by the previous
functions, which can be replaced by a historical
database for greater precision, helping Thus, the
system operator is responsible for the management
and planning of costs and powers associated with a
certain risk that is determined by the CVaR.
The commonly used probability density
functions for wind and solar energy are presented in
Section 1. Section 2 presents the formulation of the
problem taking into account the uncertainties of
costs and powers, and the conditional risk value.
The Monte Carlo method and Analytical Low-
Probability Generation Extremes are presented in
Section 3, being the simulation and case study.
Finally, Section 4 presents the conclusions due to
the results shown in the analysis.
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1.1 Normal and Log-normal PDF
In the context of analyzing solar power generation,
probability density functions are essential for
understanding the uncertainty associated with solar
irradiance and power output. [12], utilized log-
normal probability distribution functions for solar
irradiance and normal distribution functions for the
loading and unloading behavior of plug-in electric
vehicles (PEVs) to develop uncertainty cost
functions.
The article [9], develops a model with a normal
distribution (eq. 1) power, presenting the validation
for the uncertainty cost factor (UCF) by comparing
the Monte Carlo simulation with the analytical
proposal.
(1)
where is the PDF of the demand, electric vehicle
or solar power, P represents the power of the
previous variables, μ and ϕ are the mean and
standard deviation respectively of probabilistic
behavior.
Several investigations have been done to find
the probability distribution of irradiance, being the
primary resource of solar energy photovoltaic. The
function that best describes the behavior of this font
is the function Log-normal probability distribution
as [10] (eq. 2).
(2)
where is the PDF log-normal of the irradiation, G
represents the solar irradiance, λ and β are the mean
and standard deviation respectively of probabilistic
behavior.
1.2 Weibull PDF
The Weibull distribution is commonly used to
model wind speed data for wind energy
applications. The Weibull parameters, shape and
scale, can be estimated using various numerical
methods to characterize the wind resource at a given
location. Shape and scale factors are commonly
values already estimated by previous analyzes due
to historical databases for the simplicity of wind
generation studies. The application of Weibull
distribution in wind data assessment can be
extensively found, but the methods applied for
estimating the parameters still need improvement.
According to the Weibull PDF with a shape
factor (β) and scale factor (α), the wind speed
distribution can be modeled as follows, as specified
in [1], [13], [14]:
(3)
where υ is the wind speed (m/s).
In order to get the proposed uncertainty cost
functions, probability distribution functions (PDF)
of the energy primary sources are considered: log-
normal distribution for solar irradiance PDF,
Rayleigh distribution for wind speed PDF and
normal distribution for loading and unloading
behaviour PDF of electric vehicles.
2 Problem Formulation
Renewable energies that are dependent on
environmental factors, mainly such as solar
irradiation and wind speed, present variability and
uncertainty in energy production. Therefore, when
this condition occurs in energy generation, operating
costs are also related. Uncertainty costs quantify the
variability that renewable sources introduce to the
main system.
2.1 Uncertainty Powers
In the present study, it is assumed that the
generation units can be operated by companies or by
the end-users themselves, they are responsible for
both energy production and buying/selling. This
article presents an analysis of operational costs
associated with the tails of probability distribution
curves for each generation unit, specifically
focusing on power levels with lower probability of
occurrence. Depending on the probability density
function (PDF) used, these power levels are
influenced by parameters such as mean (μ) and
standard deviation (σ), along with the conditional
value-at-risk (CVaR).
2.1.1 Direct Cost and Uncertainty Solar Power
The direct cost function for the solar power plant is
determined by the following expression.
(4)
where and are the scheduled power and
direct cost coefficients of the ith solar power plant,
respectively. The constants Co and Cu represent the
costs associated with the uncertainty of solar energy.
The solar irradiance distribution can be modeled
correctly using a lognormal PDF. Using the mean
(μ) and standard deviation (σ) of the lognormal
PDF, energy conversion for solar PV is defined in
the following equation (2).
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(5)
where is the irradiance point, is the solar
irradiance in a standard environment, is the
available actual solar, and is the rated power of
the solar PV.
2.1.2 Direct Cost and Uncertainty Wind Power
The direct cost function for the wind power plant is
determined by the following expression.
(6)
where and are the scheduled power and
direct cost coefficients of the ith wind power plant,
respectively. The constants and represent the
costs associated with the uncertainty of wind
energy, where their usage is determined by the wind
power equation (2).
(7)
The wind speed distribution can be modeled
correctly using a Weibull PDF. Therefore, for the
estimation of wind power, it is determined using
scale parameter (c) and shape parameter (k) or in
real cases by historical data. For the case study,
random values determined by k and c were used.
2.2 Uncertainty Cost Formulation
To define the behavior of the generation of some
renewable source or demand in terms of probability
distribution functions, the Log-normal function,
Weibull distribution function, normal function [7],
[8], and the beta function [9], commonly related to
solar, wind generation and demand prediction,
respectively.
The proposed deterministic method is presented
to determine the penalty costs for each case,
considering the risk conditioned at 10% and 90%,
related to the cost of overestimation and
underestimation, respectively.
2.2.1 Penalty Cost due to Underestimate
Penalty costs due to underestimation depend on the
perspective of the operator. These costs arise when
the actual generation power from a renewable
source exceeds the programmed power value of the
plant, leading to energy that cannot be delivered to
the grid. Additionally, penalty costs occur when the
electric power generator, typically the main grid, is
unable to meet the energy demand of the users.
Therefore, it is essential to apply penalties to costs
associated with overestimating the available
renewable energy or failing to meet demand
requirements.
In summary, if the generating unit provides a
larger amount of actual energy than the scheduled
power (8), the surplus power may be unused, and
the grid operator is liable for the penalty cost. The
penalty cost associated with the surplus can be
referred to as follows (9):
(8)
(9)
Presenting the corresponding function for each unit
depending on the natural resource, the cost is (10):
(10)
2.2.2 Penalty Cost due to Overestimate
Due to the intermittent and uncertain nature of
primary sources (irradiation or wind) for energy
production, there is a possibility that the generating
unit will not be able to generate scheduled power. If
the actual power supplied by the renewable
generation unit is less than the scheduled power by
the operator (11), the system will require reserve
energy sources to maintain supply continuity to
consumers. Therefore, the penalty cost due to
energy scarcity should be assumed by the reserve
units and can be defined as follows (12):
(11)
(12)
Presenting the corresponding function for each unit
depending on the natural resource, the cost is (13):
(13)
2.3 Conditional Value at Risk (CVaR)
The development of an electricity system with high
penetration of energy from renewable resources
requires considerable flexibility to cover the risk of
energy curtailment and shortages, [15]. This paper
explores how the system generation portfolio of a
pool of diverse renewable sources can be
appropriately designed to balance overall planning
costs and operational flexibility constraints. The
proposed study is basically based on the production
of renewable energy with little probability
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presenting some risk when interacting with the main
grid. An index based on the conditional value at risk
(CVaR) method is introduced to quantify the risk of
any renewable source, being a parameter or
parameters necessary to determine or plan the needs
of the system, for example, the capacity or need of a
energy storage system.
The CVaR is a tool of risk measurement,
compared with the VaR (risk value), it only
considers the risk information under confidence
level, while the risk information behind the
confidence level is ignored. The CVaR measures the
average loss behind the confidence level, and the
inclusion of tail risks can better reflect the portfolio
risks. The framework of the CVaR is demonstrated
in the Figure 1.
Fig. 1: Framework of the conditional risk value
(CVaR), [16]
VaR can only determine a risk situation under
the given confidence level and doesn’t consider the
risk tail, so there are certain limitations in its
practical applications, [17].
(14)
Denote the loss function as f (x, y), where x and
y denote the probability density function of the
decision variable and the random variable,
respectively. The probability density function of y is
defined as ρ(y), then the VaR value at confidence
level is given as αβ.
(15)
The Conditional Value at Risk (CVaR) can
describe the distribution of risk outside the
confidence level. The equation (15) describes the
VaR () value and CVaR () value of the
portfolio problem, where is the CVaR value
when the loss is greater than
3 Study Case and Simulation
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. The MC method has gained
widespread acceptance for validating physical
models involving variables with associated
probability density distributions (e.g., solar
radiation) [18], [19]. Through Monte Carlo
simulation, the behavior of overestimation and
underestimation instances was studied for a
predetermined power value (Ps), considering
randomly generated values of solar irradiance and
wind speed, generated by the log-normal, normal,
and Weibull distribution functions, respectively.
The random generation of these values will be used
to obtain the generation powers of the unit in
question.
The outcome will establish values within the
generated scenarios (N=100000) for the Cost
Overestimation (Co) and Cost Underestimation
(Cu), depending on the average power value (Ps).
(16)
(17)
Equations (1), (2) and (3) represent the
probability functions that will be used for the
determination of the generated powers for each
renewable source. The obtained values were
considered for practical purposes using the MatLab
software commands "lognrnd" and "wblrnd". For
this framework, Table 1 and Table 2 shows the
initial values for the simulation were:
Table 1. PV Solar
Rated power output (Psr)
65 MW
Scheduled power (Ps)
20 MW
Solar irradiation std (Gstd)
1000 W/m2
Certain irradiation (Rc)
150 W/m2
Maximum power (Wmax)
100 MW
Mean (μ)
6
Standard deviation (σ)
0.25
After an elapsed simulation time of around
2.1644 second for solar power and 3.4711 second
for wind power, multiple statistical parameters were
obtained. It includes expected values and variances
associated with the different cost functions that were
modeled for the photovoltaic (PV) and wind turbine
(WT) generation.
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Table 2. WT Wind
Rated power output (Wsr)
150 MW
Scheduled power (Ws)
20 MW
Cut-in speed (Vi)
5 m/s
Rated output speed (Vr)
15 m/s
Cut-out speed (Vo)
45 m/s
Linear coefficient of the WT
(a, b)
(15, -75)
Rayleigh distribution scale parameter
(σ)
15.9577
Weibull Scale Parameter (√2* σ)
22.5676
Weibull Shape Parameter (k)
2
Figure 2 and Figure 3 show the results obtained
by MC for solar and wind power, respectively. The
costs of overestimation and underestimation present
a certain relationship regarding their technology or
power curves.
Fig. 2: MC simulation for Solar power
Fig. 3: MC simulation for Wind power
Figure 4 and Figure 5 show the solar and wind
power values, respectively.
Fig. 4: Solar irradiation
Fig. 5: Wind speed
3.2 Analytical Low-Probability Generation
Extremes in Solar power and Wind
Power
The equations (5) determine the power generated
about the irradiance of the place. However, CVaR
values are previously calculated (15) to determine
the risk points of interest.
The confidence level plays a very important role
in the calculation of CVaR, which determines the
minimum and maximum values of the data to be
analyzed. In this work, only the data related to
CVaR will be considered, obtaining the cost of
overestimation and underestimation determined by
the scheduled power.
For this analysis, the points of interest are
related to the management and need for energy
storage, establishing the parameters for said actions
as mentioned in section 2.3.
Figure 6 shows the CVaR at 10%, determining
the probability of energy generation on a smaller
scale, for example, it could be demonstrated that the
solar device is located in a place with cloud cover,
leading to little production and having to take into
account the need of the installation of an energy
storage system. The same occurs with the value
recorded for a CVaR at 90%, whose value would
indicate the maximum solar production of the solar
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panel, indicating that it is in a place with good solar
irradiation. The analysis of these values is
determined concerning the observer, in this case, the
network operator. The same happens with Figure 8
showing the CVaR at 10% and 90% for wind power.
Fig. 6: Solar power with CvaR 10% and 90%
Fig. 7: Power Curve of Wind Energy
The power curve (Figure 7) is related to the
wind speed, which is determined by the equations
(7). This curve presents a certain increasing
behavior due to the characteristics of the wind
turbine. Unlike the behavior for solar production,
where its generation depends on the intensity of
solar irradiation, the wind turbine will not always
generate energy even if there is a presence of wind
in the area, due to the physical characteristics of the
wind turbine. However, considering non-production
values even with the presence of the primary source,
can provide data on improvements in the system.
The histograms in Figure 6 and Figure 8 for the
solar and wind generation powers are determined by
the equations:
(18)
(19)
(20)
(21)
Each equation determines part of the structure
of the histogram determined by its probability
(counts) related by its minimum and maximum
limits (Bin). The sum of the equations (18, 19)
shows the uncertainty cost of overestimation, on the
other hand, the sum of the equations (20, 21) shows
the uncertainty cost of underestimation.
This method calculates the cost of
overestimation and underestimation in a time of
0.0551 and 0.3097 seconds for solar and wind
generation, respectively, regardless of the function
or renewable system to be used. The points of
interest are determined by the system operator with
a confidence level for the CVaR as a reference, or
by the operator himself. Table 3 and Table 4 present
the results in comparison to MC simulation.
The CVaR calculation is a risk index that
determines the expected losses that slightly exceed
the VaR. That is, expected losses greater than or
equal to VaR. In other words, risk cost for high or
minimum production, which may or may not use the
system. In the case of wind energy, given that said
production depends on wind speed, presenting
uncertainty in the primary source, the CVaR value
can coincide with the maximum production value
due to its behavior in the power curve.
The CVaR should be adjusted according to the
operator's needs or can highlight the maximum or
minimum generations for the power management or
storage capacity seen above. For practical purposes,
it was decided to have a confidence level of 0.7 for
wind energy. In summary, the value of the
confidence level represents a piece of information
that indicates the percentage of risk to be analyzed.
In the case of wind generation, the power curve is
determined by the wind speed, which regardless of
its value at that moment. If said speed is equal to or
greater than the cutting speed established by the
manufacturer or the equipment itself, the power will
be unique, presenting a certain error when
calculating the risk, since the percentile or quartile
of the operation establishes the maximum or
minimum generated. , having to adjust the
percentage value so that it presents a degree of error
in the risk by displaying a probability value.
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Fig. 8: Wind power with CvaR 10% and 30%
Table 3. Results of Solar power
MC
simulation
(sec)
Analytical
Method CVaR
Solar Power
(MW)
Error %
CVaR10
17.0025
N/A
CVaR90
40.8575
N/A
Co
13.6078
14.2643
0.0482
Cu
28.1748
28.4930
0.0113
time
1.5248
0.0551
N/A
Table 4. Results of Wind power
MC
simulation
(sec)
Analytical
Method CVaR
Wind Power
(MW)
Error
%
CVaR10
4.1161
N/A
CVaR30
148.4147
N/A
Co
99.3260
95.4860
0.0387
Cu
2453.7
2346.0
0.0439
time
2.5813
0.3097
N/A
Table 5 shows the results obtained from MC
simulation, analytical Method, and Analytical
Method CVaR using a normal function. Times
decrease significantly with the proposed methods.
Table 5. Results of Solar power with normal
function
MC
simulation
(sec)
Analytical
Method Solar
Power (MW)
Analytical
Method
CVaR Solar
Power
(MW)
CVaR10
8.1297
CVaR90
9.8788
Co
1.6025
1.6064
1.6487
Cu
1.9002
1.8774
1.9526
time
2.9160
0.0137
0.0166
4 Conclusion
Due to the uncertainty generated by renewable
sources for their production of electrical energy, the
operator of the electrical system on the part of the
main network or the part of microgrids, virtual
power plants, distributed systems, etc., must-have
tools, techniques, or methodologies. That can
minimize the risk related to the integration of these
renewable sources into any system or variable loads
in the economic dispatch of electricity, pointing to
the interaction of energy exchange. This article
shows analytical advances, which can be an
important part of the decision-making that the
operator makes every day. This proposed
mathematical formulation can be incorporated into
optimization techniques to obtain dynamic
economic dispatch models, due to the variability of
the system. The proposed analytical method, in
addition to improving the computational response
time with an error of 0.05 seconds, has the
practicality of obtaining acceptable cost values due
to overestimation and underestimation, regardless of
the probability density function that describes the
behavior of the energy production from renewable
energy sources, for example, solar and wind,
described by the Log-normal and Weibull function,
respectively. The method of this article works in a
more tactile way, due to the calculations that are
handled in the histogram graph. The proposed
method can handle the probability of a historical
database having as a reference a programmed power
about its probability. Unlike previous articles related
to the topic, this method uses the power histogram
generated from the Monte Carlo Method using
probability density functions to perform the
estimated calculation for the associated costs and
risks, instead of integral functions which have great
computational weight, however, the use of a
historical database can replace this step. The use of
the historical database provides us with greater
precision in the results obtained by using real data,
for the analysis and for practicality the Monte Carlo
Method was chosen as mentioned above.
The main objective of this work is the analysis
for the management, planning, and determination of
the use of these sources in the main network or in
any other system to be incorporated. The risk
present in the energy generated by these sources
usually indicates certain behavior due to its low
probability of generation on a smaller or larger
scale. For example, the values associated with the
powers generated through an optimization process
in a distribution system that presents a high CVaR
concerning its VaR can give us indications of the
need to have greater generation at the risk of not
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being able to cover the user demand, or on the
contrary, by presenting a relatively low CVaR
compared to its VaR, it indicates stability in the
system by being able to meet the user's needs, and
could also lend itself to the installation of a battery
bank for reserve use. Due to the high probability of
generation.
In the context of a photovoltaic system, a high
CVaR could be interpreted as the average of the
generated power that is expected to be lost in cases
where the generated power falls below the level
established by the VaR. It is a useful measure for
understanding additional risk beyond VaR and can
be used to make informed decisions about risk
management strategies.
The operator's point of view is an important
piece of information because the cost of the
overestimated/underestimated risk would depend on
the element or system to be analyzed as it is
associated with the risk that it presents in its
operation. Both the analytical method and the
proposed method about risk present very acceptable
values compared to the MC simulation, however,
the determination of risks helps the operator to
improve the system. The decision-making that the
operator, in charge of the electrical system, makes is
vital for the stability of the system itself and even
more so having a criterion of the estimated risk,
presenting the possibility of helping to better
manage the energy generated, stored, or required,
however , the use of this analysis could present
certain limitations due to the different systems being
analyzed and even more so if they are analyzed
together, as presented in the wind analysis (section
3.2), which is why the point of view of the operator
in charge of the system is of vital importance for the
formulation of the problem.
References:
[1] H. B. Huy, H. T. Doan, D. N. Vo, K. Lee, D.
Kim, Multi-objective optimal power flow of
thermal-wind-solar power system using an
adaptive geometry estimation based multi-
objective differential evolution, Applied Soft
Computing, Vol. 135, 2023, 109616. DOI:
10.1016/j.asoc.2023.110977.
[2] Z. He, J. Zhou, N. Sun, B. Jia, H. Qin,
Integrated scheduling of hydro, thermal and
wind power with spinning reserve, Energy
Procedia, Vol. 158, 2019, pp. 6302-6308.
DOI: 10.1016/j.egypro.2019.01.409.
[3] K. Majumdar, P. K. Roy, S. Banerjee,
Implementation of multi-objective chaotic
mayfly optimization for hydro-thermal-solar-
wind scheduling based on available transfer
capability problem, International
Transactions on Electrical Energy Systems,
Vol. 31, Issue 12, 2021, e13125. DOI:
10.1002/2050-7038.13029.
[4] J. Zhou, P. Lu, Y. Li, C. Wang, L. Yuan, L.
Mo, Short-term hydro-thermal-wind
complementary scheduling considering
uncertainty of wind power using an enhanced
multi-objective bee colony optimization
algorithm, Energy Conversion and
Management, Vol. 123, 2016, pp. 116-129.
DOI: 10.1016/j.enconman.2016.05.073.
[5] Z. Maa, S. Wanga, S. Lic, Y. Shi, Long-term
coordination for hydro-thermal-wind-solar
hybrid energy system of provincial power
grid, Energy Procedia, Vol. 158, 2019, pp.
6231-6235. DOI:
10.1016/j.egypro.2019.01.472.
[6] Y.V. Makarov, R. T. Guttromson, Z. Huang,
K. Subbarao, P. V. Etingov, B. B.
Chakrabarti, J. Ma, Incorporating Wind
Generation and Load Forecast Uncertainties
into Power Grid Operations, Pacific
Northwest National Laboratory, Richland,
WA, USA, 2010. DOI:
10.1109/TDC.2010.5484201.
[7] G. Pepermans, J. Driesen, D. Haeseldonckx,
R. Belmans, W. D'haeseleer, Distributed
generation: definition, benefits and issues,
Energy Policy, Vol. 33, Issue 6, 2005, pp.
787-798. DOI: 10.1016/j.enpol.2003.10.004.
[8] J. C. Arevalo, F. Santos, S. R. Rivera,
Uncertainty cost functions for solar
photovoltaic generation, wind energy
generation, and plug-in electric vehicles:
mathematical expected value and verification
by Monte Carlo simulation, International
Transactions on Electrical Energy Systems,
Vol. 29, Issue 12, 2019, e12093. DOI:
10.1504/IJPEC.2019.098620.
[9] J. L. Bernal-Agustín, R. Dufo-López,
Simulation and optimization of stand-alone
hybrid renewable energy systems, Renewable
and Sustainable Energy Reviews, Vol. 13, No.
8, 2009, pp. 2111-2118. DOI:
10.1016/j.rser.2009.01.010.
[10] M. Abido, N. Al-Ali, Multi-objective optimal
power flow using differential evolution,
Arabian Journal for Science and Engineering,
Vol. 37, No. 4, 2012, pp. 991-1005. DOI:
10.1007/s13369-012-0224-3.
[11] A. Mellit, S. A. Kalogirou, Artificial
intelligence techniques for photovoltaic
applications: A review, Progress in Energy
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.36
L. C. Perez, G. Idarraga-Ospina, S. R. Rivera
E-ISSN: 2224-350X
425
Volume 19, 2024
and Combustion Science, Vol. 34, No. 5,
2008, pp. 574-632. DOI:
10.1016/j.pecs.2008.01.001.
[12] J. Sarda, K. Pandya, K. Y. Lee, Dynamic
optimal power flow with cross entropy
covariance matrix adaption evolutionary
strategy for systems with electric vehicles and
renewable generators, International Journal
of Energy Research, Vol. 45, No. 7, 2021, pp.
10869–10881. DOI: 10.1002/er.6571.
[13] M. Gul, N. Tai, W. Huang, M. H. Nadeem, M.
Yu, Evaluation of Wind Energy Potential
Using an Optimum Approach based on
Maximum Distance Metric, Sustainability,
Vol. 12, No. 5, 2020, 1745. DOI:
10.3390/su12051999.
[14] A. K. Khamees, A. Y. Abdelaziz, M. R.
Eskaros, M. A. Attia, A. O. Badr, The
Mixture of Probability Distribution Functions
for Wind and Photovoltaic Power Systems
Using a Metaheuristic Method, Processes,
Vol. 10, No. 11, 2022, 255. DOI:
10.3390/pr10112446.
[15] M. Du, Y. Huang, J. Liu, G. Wu, S. Jawad,
CVaR-based generation expansion planning
of cascaded hydro-photovoltaic-pumped
storage system with uncertain solar power
considering flexibility constraints, IET
Generation, Transmission & Distribution,
Vol. 15, No. 14, 2021, pp. 2953-2966. DOI:
10.1049/gtd2.12232.
[16] R. Gao, H. Guo, R. Zhang, T. Mao, Q. Xu, B.
Zhou, P. Yang, A Two-Stage Dispatch
Mechanism for Virtual Power Plant Utilizing
the CVaR Theory in the Electricity Spot
Market, Energies, Vol. 12, No. 17, 2019,
3402. DOI: 10.3390/en12173402.
[17] L. Ju, P. Li, Q. Tan, Z. Tan, G. De, A CVaR-
Robust Risk Aversion Scheduling Model for
Virtual Power Plants Connected with Wind-
Photovoltaic-Hydropower-Energy Storage
Systems, Conventional Gas Turbines and
Incentive-Based Demand Responses,
Energies, Vol. 11, No. 11, 2018, 2903. DOI:
10.3390/en11112903.
[18] M. M. Ghahderijani, S. M. Barakati, A.
Jamshidi, Application of Stochastic
Simulation Method in Reliability Assessment
of a PV-Wind-Diesel-SOFC Hybrid
Microgrid, International Journal of
Engineering and Technology, Vol. 4, No. 5,
2012, pp. 586-589. DOI:
10.7763/IJET.2012.V4.438.
[19] A. A. Téllez, L. Ortiz, M. Ruiz, K. Narayanan
and S. Varela, Optimal Location of Reclosers
in Electrical Distribution Systems Considering
Multicriteria Decision Through the
Generation of Scenarios Using the Montecarlo
Method, IEEE Access, Vol. 11, 2023, pp.
68853-68871. DOI:
10.1109/ACCESS.2023.3252411.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Conceptualization L.C.P. and S.R.R.; methodology,
L.C.P. and S.R.R.; software, L.C.P.; validation,
L.C.P. and S.R.R.; formal analysis, L.C.P. and
S.R.R.; investigation, S.R.R.; resources, L.C.P. and
S.R.R.; data curation, L.C.P and S.R.R.; writing—
original draft preparation, L.C.P. and S.R.R.;
writing—review and editing, L.C.P.; 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.
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
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.36
L. C. Perez, G. Idarraga-Ospina, S. R. Rivera
E-ISSN: 2224-350X
426
Volume 19, 2024
