Simplified Approach for Wind Uncertainty Cost Functions using a
Mixture of Uniform Probability Distribution
LIBARDO ACERO GARCÍA1, MUHAMMAD ATIQ UR REHMAN2,
SERGIO RAUL RIVERA RODRIGUEZ1
1Electrical and Electronics Engineering Department and CREG,
Universidad Nacional de Colombia,
Carrera 30 Número 45-03, Bogota,
COLOMBIA
2Electrical and Electronics Engineering Department,
International Islamic University, Islamabad,
PAKISTAN
Abstract: - Wind energy generation is an individual source of energy considered in renewable energy resources.
It is completely dependent on the natural process of air blown in the area or past forecasting of air blown data.
It is considered an uncertain source of energy and less reliable in the context of real-time energy provisions. To
make this wind energy more reliable and consistent, we are introducing a simplified approach for wind
uncertainty cost functions based on uniform probability distributions and its evaluation for usage. We have
developed analytical mathematics cost functions based on several assumptions and they are constructed using
Weibull and Rayleigh probability distributions. The proposed method produces the required results to make this
energy reliable and consistent with the distribution networks. Computing time and complexity are less as
compared to the other methods to minimize the uncertainty of wind energy.
Key-Words: - Microgrids, Renewable Energy Resources, Stochastic Processes, Wind Energy, Probability
Distribution, Uncertainty Cost Functions, Weibull and Rayleigh Probability Distributions.
Received: February 28, 2023. Revised: December 13, 2023. Accepted: December 23, 2023. Published: March 12, 2024.
1 Introduction
Wind energy is an integral part of renewable energy
resources having environmental impacts as
compared to traditional energy resources. Wind
energy resources are considered uncertain in
behavior producing the factor of a stochastic
process. To handle this uncertainty behavior, we
need to design and develop appropriate primary
methods for variable uncertainty costs, [1]. The
distribution system networks have vast pressure to
bear such uncertain behavior of wind energies. The
exhaustive search approach is most relevant in this
situation to see the iterative solutions and
visualization based on several possibilities available,
[2]. In this regard, the optimal power flow is
required to improve the uncertainties in wind power
generation and make it more reliable and available
to the end customers. A constrained handling
approach can be used for this purpose rather than a
cost function handling approach, [3]. Being an
uncertain energy generation from wind, there is a
vast impact on the charging infrastructure of electric
and hybrid electric vehicles.
This uncertain energy generation can affect the
power system strategies to operate it properly daily.
So, we can use such strategies which are based on
time of use energy and demand profile, [4]. The
hierarchical coordinated charging system can
improve the uncertain power produced by wind. The
primary transformer is operating to minimize the
energy cost in such situations where induction of
energy in the power system is done near the end
customer metering system. The distribution system
operator is responsible for deciding where to start
the charging system or to delay its available
feasibility, [5]. Smart grids are playing an important
role in developing and delivering to handle
uncertain powers to end users, but currently, smart
grids are relying on renewable energy resources.
Smart grids have the capacity and capability to
handle the uncertain power in normal and
emergencies. Combinatorial strategy has the
potential to cater the several renewable energy
resources to optimize the supply and demand of
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.7
Libardo Acero García, Muhammad Atiq Ur Rehman,
Sergio Raul Rivera Rodriguez
E-ISSN: 2224-350X
54
Volume 19, 2024
power, [6]. Energy storage systems can also be the
first choice to mitigate the uncertainty of wind
energy generation in recent times. It improves the
microgrid's scheduling and dispatch of electricity. It
also improves the operation of power systems
making the grid smart in power outage of several
resources. In this regard, the energy storage systems'
useful life can have an important impact to decrease
the cost of the system and reliability, [7].
The reliability and dispatch of power vary when
using renewable energy sources, such as wind
generation. We can minimize the variation cost by
using the interior point methodology to make
renewable energy availability linear, [8]. The real-
time control of the operation of microgrids having
renewable energy resources is considered an
important problem for making the grid; smart. The
energy produced from uncertain renewable energy
resources can be challenging for the market prices
of all stakeholders involved. In this regard, the
strategy of parallel computing can provide the
maximum power to the market for clearing
transactions and prices, [9]. The supply of power
needs more reliability and less uncertainty by the
distribution companies to create a balance in
economic and technical benefits. The fault location
and fault tolerance must be precisely catered for in
real time to increase the reliability of the power
system. To enhance the reliability of the power
distribution networks, we need to do the reliability
assessment including computing time and
improvements, [10].
The quality and reliability of the distribution
networks are dependent on the solution of sequential
network failure or reduction in dis-connectivity. For
this purpose, Markov's decision process in
predicting the failure can be identified. Sequential
attacks or disconnection from the distribution
network or transmission network can be
strategically improved by reward functions by using
state transition probabilities, [11]. The distribution
networks are more complex while talking about the
control of their operations. The distribution control
strategies are more convenient and robust to handle
network operations and variations. Due to the
continuous variation behavior of microgrids,
dynamic population games can handle variations
precisely and maintain the frequency of the group of
microgrids, [12].
Currently, the virtual concept of controlling and
monitoring the power systems is in discussion. We
can develop virtual systems based on several
strategies for optimal results rather than real
physical systems. Such methods upon evaluation
can give optimal results for the power system’s
performance, [13]. The energy policies and
strategies are formulated globally to test and
evaluate the existing energy resources with several
existing tools. The Long-range Energy Alternatives
Planning (LEAP) system tool can be used to
compare and measure the performance of alternative
fuels by evaluating their costs and energy generation
capacities, [14]. The uncertainty of wind energy in
the power systems can affect the electric and hybrid
electric vehicles charging conditions. We need to
optimize the power provided by the distribution
systems operator and to meet the load curves
produced by the vehicles. An optimal power profile
can be designed to address the uncertainties
produced by the wind energy vulnerability, [15].
Based on the previous literature in section I and
assumptions made in section II, we have developed
the analytical cost functions to deal with the
uncertain behavior of wind energy generation.
2 Problem Formulation: Wind
Available Power through Three
Uniform Distributions
The Weibull and Rayleigh probability distributions
are used for modeling the wind speed. Using the
power and primary source curve of Eolic
technology, it is possible to get the available power
from the Eolic generation process to present it as a
mixture of three probability distributions. Thus, the
available power histogram can be well described by
three uniform distributions for different parameters
of the Weibull distribution, namely the scale factor
(c) and shape factor (k). Additionally, it is well
known that the Rayleigh distribution can be viewed
as a special case of the Weibull distribution where
the shape factor (k) is known to equal 2.
Figure 1(a) data is obtained by setting the
parameters shown in Table 1(a) for the power
histogram not limited to 3 uniform distributions
from the wind speed histogram.
Table 1(a). Weibull- Rayleigh parameters values
Shape factor
(k)
Scale factor (c)
Shape factor
(k)
15.9577
sqrt (2) *sg
2
The Figure 1(b) data is obtained by setting the
parameters shown in Table 1(b) for power
histogram limited to 3 uniform distributions from
wind speed histogram.
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.7
Libardo Acero García, Muhammad Atiq Ur Rehman,
Sergio Raul Rivera Rodriguez
E-ISSN: 2224-350X
55
Volume 19, 2024
Table 1(b). Weibull- Rayleigh parameters values
Shape factor
(k)
Scale factor (c)
Shape factor
(k)
15.9577
sqrt (2) *sg
2
The Figure 2(a) data is obtained by setting the
parameters shown in Table 2(a) for power histogram
not limited to 3 uniform distributions from wind
speed histogram.
Table 2(a). Weibull- Rayleigh parameters values
Shape factor
(k)
Scale factor (c)
Shape factor
(k)
8
sqrt (2) *sg
2
The Figure 2(b) data is obtained by setting the
parameters shown in Table 2(b) for power
histogram limited to 3 uniform distributions from
wind speed histogram.
Table 2(b). Weibull- Rayleigh parameters values
Shape factor
(k)
Scale factor (c)
Shape factor
(k)
8
sqrt (2) *sg
2
The Figure 3(a) data is obtained by setting the
parameters shown in Table 3(a) for power histogram
not limited to 3 uniform distributions from wind
speed histogram.
Fig. 1(a): Power histogram not limited to 3 uniform
distributions
Fig. 1(b): Power histogram limited to 3 uniform
distributions.
Table 3(a). Weibull parameters values
Shape factor
(k)
Scale factor (c)
Shape factor
(k)
16
sqrt (2) *sg
4
The Figure 3(b) data is obtained by setting the
parameters shown in Table 3(b) for power
histogram limited to 3 uniform distributions from
wind speed histogram.
Table 3(b). Weibull parameters values
Shape factor
(k)
Scale factor (c)
Shape factor
(k)
16
sqrt (2) *sg
4
Fig. 2(a): Power histogram not limited to 3 uniform
distributions
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.7
Libardo Acero García, Muhammad Atiq Ur Rehman,
Sergio Raul Rivera Rodriguez
E-ISSN: 2224-350X
56
Volume 19, 2024
Fig. 2 (b): Power histogram limited to 3 uniform
distributions
3 Problem Solution: Analytical
Development for Wind Uncertainty
Cost Functions
3.1 Power Histogram Description
The available power (P) can be described by using
the power histogram described in the previous
section. The scheduled power to be handled by the
operator can be categorized into three regions as
follows:
Case A; Region I: Ps is less than b and
bigger than a
Fig. 3(a): Power histogram not limited to 3 uniform
distributions
Fig. 3(b): Power histogram limited to 3 uniform
distributions
Case B; Region II: Ps is less than c and
bigger than b.
Case C; Region III: Ps is less than d and
bigger than c.
Additionally, each uniform distribution will have
a ponderation w1, w2, and w3, respectively for each
region, as depicted in Figure 4.
Fig. 4: Available wind power histogram limited to 3
uniform distributions
To develop a mathematical formulation of
uncertainty cost functions for wind energy
resources, the wind uncertainty cost function must
be in terms of , that is to say . It can have
the following three cases based on the analytical
development of a mixture of three probability
distributions.
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.7
Libardo Acero García, Muhammad Atiq Ur Rehman,
Sergio Raul Rivera Rodriguez
E-ISSN: 2224-350X
57
Volume 19, 2024
3.2 Analytical Development of UCFs for the
Overestimation Part
For wind uncertainty cost function for
overestimation part , the following cases can
be formulated mathematically ( is the cost for
using a back storage system in controllable
renewable sources, used to inject the power
difference from the available power P to the
scheduled power ):
Case A:
If then,
(Ao-1)
Case B:
If then,
(Bo-1)
Case C:
If then,
(Co-1)
The analytical development for wind uncertainty
cost function for overestimation part as
follows:
Case A:
=
=
(Ao-2)
Case B:
+
=
+
=
+
=
(Bo-2)
Case C:
+
+
=
+
(Co-2)
3.3 Analytical Development of UCFs for the
Underestimation Part
For wind uncertainty cost function for
underestimation part , the following cases can
be formulated mathematically ( is the cost for
storage energy in controllable renewable sources,
used to storage the power difference from the
scheduled power to the available power P):
Case A:
If then,
(Au-3)
Case B:
If then.,
(Bu-3)
Case C:
If then,
(Cu-3)
In this way, the analytical development for wind
uncertainty cost function for underestimation part
as follows:
Case A:
+
+
=
+
+
=
+
+
(Au-4)
Case B:
+
=
+
=
(Bu-4)
Case C:
=
=
(Cu-4)
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.7
Libardo Acero García, Muhammad Atiq Ur Rehman,
Sergio Raul Rivera Rodriguez
E-ISSN: 2224-350X
58
Volume 19, 2024
4 Simulation and Validation
In Figure 5, the Monte Carlo process used original
simulation results based on Weibull wind speed and
uncertainty cost histograms, and they are shown. A
power histogram limited to 3 uniform distributions
is used to find out the cost due to overestimating or
cost due to underestimating scenarios. Similarly, the
other simulation results showed the injected power
in the form of a mixture of three uniform
distributions as in Figure 1(b), Figure 2(b) and
Figure 3(b) also show similar results for cost due to
overestimate or cost due to underestimate scenarios.
By considering the overestimation and
underestimation part (both cases), a total uncertainty
cost functions histogram is shown in Figure 6.
Fig. 5: Weibull wind speed and uncertainty cost
Fig. 6: UCF histogram
Similarly, we can find out the power histograms
of other injected power in the form of a mixture of
three uniform distributions. The Montecarlo
Expected cost of the UCF is 2.0858e+03 $, using
the Weibull distribution for the simulations. The
analytical value using the analytical formulation
presented in [5], is in the form of the following
equations:
(1)
(2)
Applying the complex equation (1) and (2), the
analytical uncertainty cost functions is 2.0847e+03
$. Using the proposed formulation in the previous
section, we get the following UCF 2.1536e+03 $
which is an error of 3.3 %. Finally, we developed
the Montecarlo simulation with the three uniform
distributions and we got the following UCF
2.1548e+03 $, it is an error of 5.7510e-02 %
between the Montecarlo simulation and the
analytical expression of the previous section. That is
to say, if we do a variation of the whole possible
scheduled power, we can get:
i. The whole analytical method (equations
very complex, (1) and (2)
ii. The proposed method (simple equations)
iii. The Montecarlo simulation (In this
method the process computing time is so
expensive).
5 Conclusion
Wind energy generation produces uncertain power
to the distribution networks and has stochastic
behavior. We can deal with such uncertain behavior
of wind energy generation by designing appropriate
methods to handle uncertain costs as well. The
produced power histograms are described by three
uniform distributions and are used to develop
uncertainty cost functions for making the energy
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.7
Libardo Acero García, Muhammad Atiq Ur Rehman,
Sergio Raul Rivera Rodriguez
E-ISSN: 2224-350X
59
Volume 19, 2024
linear. We have used a simplified approach for wind
uncertainty cost functions based on a mixture of
uniform probability distribution producing required
results to make linear and reliable energy
availability. The virtual inertia on distribution
system networks and the uncertain behavior of wind
energies are minimized by using this approach.
Optimal power flow is improved in this proposed
methodology making the wind energy more reliable
and consistent for end users.
References:
[1] C. E. Tungom, B. Niu, and H. Wang,
“Hierarchical framework for demand
prediction and iterative optimization of EV
charging network infrastructure under
uncertainty with cost and quality-of-service
consideration,” Expert Syst Appl, vol. 237,
p. 121761, Mar. 2024, doi:
10.1016/J.ESWA.2023.121761.
[2] David Gregoratti, Javier Matamoros.
"Distributed Energy Trading: The Multiple-
Microgrid Case." IEEE Transactions on
Industrial Electronics, 62, 4 (2015): 2551-
2559
[3] T. Valencia-Zuluaga, S. Rivera, “A Fast
Decomposition Method to Solve a Security-
Constrained Optimal Power Flow (SCOPF)
Problem through Constraint Handling,”
IEEE Access, vol. 9, no. March, pp. 52812–
52824, 2021, doi:
10.1109/ACCESS.2021.3067206.
[4] D. M. Gado, I. Hamdan, S. Kamel, A. Y.
Abdelaziz, and F. Jurado, “Optimizing
Energy Consumption in Smart Homes: A
Comprehensive Review of Demand Side
Management Strategies,” pp. 1–10, Feb.
2024, 2023 IEEE CHILEAN Conference on
Electrical, Electronics Engineering,
Information and Communication
Technologies (CHILECON), doi:
10.1109/CHILECON60335.2023.10418744.
[5] A. M. Sanchez, G. E. Coria, A. A. Romero,
and S. R. Rivera, “An improved
methodology for the hierarchical
coordination of PEV Charging,” IEEE
Access, vol. 7, no. November, pp. 141754–
141765, 2019, doi:
10.1109/ACCESS.2019.2943295.
[6] Y. Li, “Adjustable Capability Evaluation of
Integrated Energy Systems Considering
Demand Response and Economic
Constraints,” Energies, (Basel), vol. 16, no.
24, Dec. 2023, doi: 10.3390/en16248048.
[7] C. Baron, A. S. Al-Sumaiti, and S. Rivera,
“Impact of energy storage useful life on
intelligent microgrid scheduling,” Energies,
(Basel), vol. 13, no. 4, 2020, doi:
10.3390/en13040957.
[8] V. Mihaly, M. Stanese, M. Susca, and P.
Dobra, “Interior Point Methods for
Renewable Energy Management,” 2020
22nd IEEE International Conference on
Automation, Quality and Testing, Robotics -
THETA, AQTR 2020 - Proceedings, May
2020, doi:
10.1109/AQTR49680.2020.9129953.
[9] S. M. A. N. Javareshk, Z. Safari, M.
Niaemanesh, J. Mohammadzade, and H.
Shafahi, “Optimal Bidding Strategy for
Purchasing and Selling the Electricity by the
EV Aggregator in the Energy Market, Based
on Actor-Critic Algorithm,” 2023 27th
International Electrical Power Distribution
Conference, EPDC 2023, pp. 53–59, 2023,
doi: 10.1109/EPDC59105.2023.10218862.
[10] D. Midence, S. Rivera, and A. Vargas,
“Reliability assessment in power
distribution networks by logical and matrix
operations,” 2008 IEEE/PES Transmission
and Distribution Conference and
Exposition: Latin America, T and D-LA, no.
September 2008, 2008, doi: 10.1109/TDC-
LA.2008.4641696.
[11] C. Caro-Ruiz, A. S. Al-Sumaiti, S. Rivera,
and E. Mojica-Nava, “A MDP-Based
Vulnerability Analysis of Power Networks
Considering Network Topology and
Transmission Capacity,” IEEE Access, vol.
8, pp. 2032–2041, 2020, doi:
10.1109/ACCESS.2019.2962139.
[12] E. Mojica-Nava, S. Rivera, and N. Quijano,
“Distributed dispatch control in microgrids
with network losses,” 2016 IEEE
Conference on Control Applications, CCA
2016, no. March 2018, pp. 285–290, 2016,
doi: 10.1109/CCA.2016.7587850.
[13] J. Ramírez-Romero, D. Rodriguez, and S.
Rivera, “Teaching using a synchronous
machine virtual laboratory,” Global Journal
of Engineering Education, vol. 22, no. 2, pp.
123–129, 2020.
[14] W. C. Flores, B. Bustamante, H. N. Pino, A.
Al-Sumaiti, and S. Rivera, “A national
strategy proposal for improved cooking
stove adoption in Honduras: Energy
consumption and cost-benefit analysis,”
Energies, (Basel), vol. 13, no. 4, 2020, doi:
10.3390/en13040921.
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.7
Libardo Acero García, Muhammad Atiq Ur Rehman,
Sergio Raul Rivera Rodriguez
E-ISSN: 2224-350X
60
Volume 19, 2024
[15] H. Bitaraf and S. Rahman, “Optimal
operation of energy storage to minimize
wind spillage and mitigate wind power
forecast errors,” IEEE Power and Energy
Society General Meeting, vol. 2016-
November, Nov. 2016, doi:
10.1109/PESGM.2016.7741550.
Contribution of Individual Authors to the
Creation of a Scientific Article
- L. Acero and M. Rehman carried out the
simulation and the optimization.
- S. Rivera has implemented the Montecarlo
Algorithm
- L. Acero, M. Rehman and S. Rivera were
responsible for the Statistics.
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
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
_US
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.7
Libardo Acero García, Muhammad Atiq Ur Rehman,
Sergio Raul Rivera Rodriguez
E-ISSN: 2224-350X
61
Volume 19, 2024
