Abstract: - Navigating the secheduling of generation resources of energy in power systems marked by a

signicant presence of renewable generation involves intricate optimization challenges. The conventional

tools for resolving such challenges include programming techniques and heuristic approaches, both contin-

gent upon a precisely articulated target function for optimization. Traditional optimization tools rely on

precisely dened target functions, but the evolving landscape of power systems introduces complexity, espe-

cially with unpredictable behaviors of renewable sources. The research specically quanties penalty costs

associated with photovoltaic (PV) generators, employing probabilistic methods for a robust mathematical

analysis. The developed analytical model enhances adaptability in economic dispatch problems, considering

uncertainty in decision-making. Validation using Monte Carlo simulation emphasizes uncertainty in PV

generation and highlights the advantages of the proposed analytic model. The quadratic form of the model

aligns coherently with simulation outcomes, contributing signicantly to understanding uncertainty quan-

tication in solar power modeling. The research aims to rene controllable solar power models, establish

robust uncertainty cost functions, and improve the accuracy of economic dispatch strategies. Ultimately,

this work promotes the seamless integration of solar energy into diverse and dynamic energy grids.

Key-Words: - Controllable Renewable Generation, Montecarlo Simulations, Power Systems, Uncertainty

Quantication, Solar Power

Received: March 15, 2023. Revised: December 27, 2023. Accepted: February 13, 2024. Published: March 27, 2024.

1 Introduction

The need for advanced modeling techniques

in the energy dispatch optimization in power

systems, particularly in the context of a substan-

tial presence of renewable generation, is due to

uncertainty quantication approaches in order to

solve this optimization problem, [1]. Heuristic

algorithms have served as tools to address these

uncertainty complexities, in order to get the

a robust energy dispatch, [2]. In these issues is

relevant to both relying on precisely dened target

functions for optimization and using uncertainty

model techniques for handling the variability of

renewable primary sources (wind speed and solar

irradiation), [3]. The modern power systems in

need of trace of uncertainty, with high penetration

of renewables, are characterized by complexity

due to variability of the sources with inherently

unpredictable behaviors. In this way, the power

system operator are in the need of having robust

modeling strategies, [2]. Thus, this study and

mathametical proposed approach intend to delve

into the intricacies of controllable solar power

technologies (solar generators with back-ups for

guaranteed a range of power in a time instance),

specically focusing on the quantication of

Quantification of Uncertainty Cost Functions for Controllable Solar

Power Modeling

SERGIO RAUL RIVERA RODRIGUEZ

Universidad Nacional de Colombia

Department of Electrical Engineering

Carrera 30 No. 45-03 Bogota

COLOMBIA

AMEENA AL-SUMAITI

Khalifa University

Electrical Engineering Department

Advanced Power and Energy Center

P.O. Box: 127788, Abu Dhabi

UAE

TAREEFA S. ALSUMAITI

United Arab Emirates University

Geography and Urban Sustainability Department

P.O. Box: 127788, Abu Dhabi

UAE

WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.11

Sergio Raul Rivera Rodriguez,

Ameena Al-Sumaiti, Tareefa S. Alsumaiti

E-ISSN: 2224-350X

Volume 19, 2024

penalty costs associated with photovoltaic (PV)

generators in two conditions, underestimation and

overestimation of the available power, [3].

A review of the existing literature and high pene-

tartion of solar technologies in several countries,

reveals a increasing recognition of the challenges

posed by uncertainty in renewable energy systems

with a big capacity inside of the power systems,

particularly in the economic dispatch of power.

Traditional optimization techniques have been

employed, often relying on deterministic models

that struggle to capture the stochastic nature

of renewable resources, [4]. In response to this

need, our study proposes a rigorous mathematical

analysis that leverages the uses of probabilistic

methods, aiming a contrast to traditional deter-

ministic models, [5].

The importance of incorporating uncertainty

quantication into modeling frameworks are

recent advancements highlights, emphasizing the

need for robust cost functions to enhance the

accuracy of economic dispatch strategies, [6].

A comprehensive review of literature in the

adaptability and robustness indicates a need of

shift toward a paradigm integrating controllable

renewable systems into economic dispatch target

functions, [7], [8]. This shift aims to enhance the

operator process for decision-making, considering

the evolving nature of power systems with the

inclusion of renewable sources, [8].

To validate our proposed approach, based in

previous developments ( [9], [10], [11], [12]), we

draw upon Monte Carlo simulation techniques,

emphasizing the uncertainty associated with PV

generation, especially in the context of the whole

posiibilities capacity of energy storage, [9], [10].

The proposed formulation is a new development in

simulation-based validation methodologies, with

an emphasis on the importance of aligning analyt-

ical frameworks with simulation outcomes, [11].

Our study presented in this paper contributes to

this discourse by presenting a novel analytical

model, grounded in a uniform power distribution

(it is an extension of the study presented in [12],

with a variation of the ranges of the scheduled

power), and validating its performance against

Monte Carlo simulations. In this way, section

2 presents the analytical development; section 3

depicts the validation and application calculating

the energy to be stored to balance demand and

solar generation, and section 4 draws a discussion

and conclusion.

2 Controllable Photo-voltaic Cost

Function-Analytical Development

The mathematical uncertainty cost functions

considering uniform distributions for solar gener-

ators are derived considering a scheduled power

(Ps) in a determined range, normally from an ar-

bitary maximum and minimum value. In [12], it is

considered a range between a minimum power and

a maximum power, assuming that the probability

density function for the available generated power

f[P](from the technology used to convert primary

source (solar irradiation) in electric power) is de-

ned with an uniform distribution:

f[P] = 





Pmax −Pmin for Pmin ≤P≤Pmax,

0for P < Pmin or P > Pmax

(1)

It is used a linear function in order to handle

the penalty cost due to an underestimation y=

Cu[P] = Cu(P−Ps)(the same would be in the

overestimation case). In this way, it is possible to

determine the corresponding expected penalty cost

function as follows:

E[y] = Z∞

−∞

yf (y)dy (2)

→E[Cu(P)] = Cu

Pmax −Pmin P2

2−

PsPmax +P2

max

2(3)

On the other hand, the expected cost func-

tion for the overestimation with z=Co[P] =

Co(Ps−P)for controllable solar generation can

be obtained:

E[z] = Z∞

−∞

zf (z)dz (4)

→E[Co(P)] = Co

Pmax −Pmin P2

2−

PsPmin +P2

min

2(5)

The previous results (from [12]) make it pos-

sible to calculate the expected uncertainty cost

function (UCF), which describes a remarkable

quadratic pattern, something useful for conven-

tional economic dispatch softwares.

→E[UCF ] = E[Cu(P)] + E[Co(P)] (6)

The need to broaden and continue develop-

ment of this framework of analysis come from the

WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.11

Sergio Raul Rivera Rodriguez,

Ameena Al-Sumaiti, Tareefa S. Alsumaiti

E-ISSN: 2224-350X

Volume 19, 2024

understanding that scheduled power is complex.

Recognizing that scheduled power (Ps) is not a

single variable but rather occurs in a variety of op-

erational states, the study presented in this section

intends to improve our comprehension by group-

ing Psinto three dierent areas (in [12], was used

only one area). The study conducted extends be-

yond the conventional distinction between Pmin

and Pmax to encompass areas where Psis (i) less

than Pmin, (ii) between Pmin and Pmax, and (iii)

beyond Pmax but less than Pmin plus the back Bat-

tery Capacity Pbatterycapacity .

In this way, we expanded the work presented

in [12], in the following way:

• We consider that the scheduled power (Ps) not

only can vary from Pmin to Pmax, but instead

the scheduled power could be in three dierent

regions:

Region 1, Ps< Pmin

Region 2, Pmin < Ps< Pmax

Region 3, Pmax < Ps<(Pmin +Pbatterycapacity)

In this new development, we update the equa-

tions to be presented in three regions:

• One equation (there is not overestimation con-

dition) for Region 1 f1: For this region in

which Ps is less than Pmin, the UCF is de-

noted by:

f1(Ps) = E[UCF]→where{Ps≤Pmin

P∈(Pmin, Pmin)

(7)

E[UCF] = ZPmax

Pmin

(Cu(P−Ps)I(P > Ps)) dP

(8)

=Cu

Pmax −Pmin ZPmax

Pmin

(P−Ps)dP (9)

=Cu

Pmax −Pmin

(Ps

2



Pmax

Pmin

−PsPmax



Pmax

Pmin

)(10)

=Cu

Pmax −Pmin

(P2

max

2−P2

min

2−PsPmax +PsPmax)

(11)

=Cu

Pmax −Pmin (Pmax +Pmin)(Pmax −Pmin)

−PsPmax +PsPmin)

(12)

=Cu(Pmax +Pmin

2−Ps)(13)

• Two equations (overestimation and understi-

mation conditions) for Region 2 f2:

f2(Ps) = E[UCF]→where{Pmin ≤Ps≤Pmax

P∈(Pmin, Pmax)

(14)

=Co

Pmax −Pmin

(P2

2−PsPmax +P2

max

2)(15)

+Cu

Pmax −Pmin

(P2

2−PsPmin +P2

min

2)(16)

this is from

Equation A:

E[Co] = ZPmax

Pmin

(Co(Ps−P)I(P < Ps)) dP (17)

=Co

Pmax −Pmin

(Ps.P 



Pmin

−P2

2



Pmin

)(18)

=Co

Pmax −Pmin

(P2

s−P2

s,min −P2

2−P2

min

2)(19)

=Co

Pmax −Pmin

(P2

2−PsPmin −P2

min

2)(20)

Equation B:

E[Cu] = CuZPmax

Pmin

(P−Ps)(I(P > Ps)) dP

(21)

=Cu

Pmax −Pmin ZPmax

(P−Ps)dP (22)

=Cu

Pmax −Pmin

(P2

2



Pmax

−Ps.P 



Pmax

)

=Cu

Pmax −Pmin

(P2

max

2−P2

2−PsPmax +P2

=Cu

Pmax −Pmin

(P2

2−PsPmax −P2

max

2)(23)

WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.11

Sergio Raul Rivera Rodriguez,

Ameena Al-Sumaiti, Tareefa S. Alsumaiti

E-ISSN: 2224-350X

Volume 19, 2024

• One equation (there is not underestimate

condition) for Region 3 f3:

f3(Ps) = E[UCF]→where

{Pmax ≤Ps≤Pmin +Pbatterycapacity

P∈(Pmin, Pmax)

(24)

E[Co] = ZPmax

Pmin

(Co(Ps−P)I(P < Ps)) dP (25)

=ZPmax

Pmin

(Ps−P)dP (26)

=Co

Pmax −Pmin

(Ps.P 



Pmax

Pmin

−P2

2



Pmax

Pmin

)(27)

=Co

Pmax −Pmin

((PsPmax −PsPmin)−P2

max

2+P2

min

(28)

=Co

Pmax −Pmin

(Ps(Pmax −PsPmin)−P2

max

2+P2

min

=Co(Ps−Pmax +Pmin

2)(29)

3 Validation and Application:

Energy to be stored to balance

demand and solar generation

If the uncertainty cost functions (developed in

the previous section), which has the following

units ($/hour), is multiplied by the system fac-

tor (k) that represents the inverse of the energy

cost (kWh/$), we obtain the expected value of the

power injected by the solar panels (this expected

value considers the probability distribution of the

solar power value available in the panels). This

study oers a baseline for assessing the nancial

eects of solar power generation by concentrating

on the scheduled scenario and accounting for both

the system factor (k) and uncertainty cost func-

tions. In this way, it is possible to calculate the

energy to be stored (Eb) to balance power demand

(Pdemand) and solar generation (Psolar) in a sys-

tem:

Eb=−Z24

Pdemanddt +Z24

Psolardt

where

Eb=−Z24

Pdemanddt +Z24

E[UCF]∗kdt

It is possible to get the E[UCF ]and kfrom

the previous section (analytical method) and with

Montecarlo simulations. That is to say, we used

a random number generator that follows a uni-

form distribution between a minimum and max-

imum value for each time instance. The Monte

Carlo simulations descriptions can be described in

5 steps:

i) Solar power (green: schedule power, blue:

min, red:max)

The three main accounts situations maximum

power (red), scheduled power (green) and mini-

mum power (blue) are taken into consideration.

Considering the predictive modeling or operator

scheduling data, the scheduled power is displayed

by the green curve in the simulation. We present

two possible patters (Figure 1 and Figure 2) for

scheduling (green curve): A) mean value between

Pmin and Pmax; and B) an arbitary pattern. The

lower limits of solar power generation in our calcu-

lations are shown by the blue curve. This scenario

takes into consideration times when there is less

sunlight or unanticipated changes that cause so-

lar panels to produce the least amount of power.

The maximum solar power generation limits in

our Monte Carlo simulations are shown by the

red curve. In this scenario, ideal circumstances

are taken into account, where solar panels provide

power to the fullest extent possible.

Fig. 1: Solar Power (pattern A).

ii) Monte Carlo scenarios between max and

min values The study explores the complex dy-

namics between the maximum (red) and minimum

(blue) values of solar power generation in these

WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.11

Sergio Raul Rivera Rodriguez,

Ameena Al-Sumaiti, Tareefa S. Alsumaiti

E-ISSN: 2224-350X

Volume 19, 2024

Fig. 2: Solar Power (pattern B).

Monte Carlo simulations. The study captures the

subtle dierences in solar power generation and

their associated economic repercussions by inves-

tigating scenarios within this range. The Monte

Carlo simulations carried out inside the dynamic

range, in contrast to the discrete average, mini-

mum, and maximum scenarios, take into account

a spectrum of solar power values between the max-

imum (red) and minimum (blue) criteria (Figure

3).

Fig. 3: Montecario scenarios between Max and

Min values.

iii) Random Escenario of Solar generation The

use of random solar generation scenarios in Monte

Carlo simulations has led to a more realistic and

exible method of modeling controlled solar elec-

tricity, [3]. Decision-makers can increase the de-

pendability and nancial sustainability of renew-

able energy systems by accepting and valuing

the unpredictability of solar power generation, [4].

This methodology generates completely random

situations, embracing the stochastic character of

solar power generation instead of focusing on spec-

ied average, minimum, or maximum values (Fig-

ure 4).

Fig. 4: Random scenario of solar generation.

iv) Expected energy from Montecarlo scenar-

ios When computing predicted energy in Monte

Carlo simulations of solar power generation, one

must estimate the average or mean power out-

put value over a number of simulation iterations,

and it is possible to get the histogram (Figure 5).

Because the model is stochastic, every time the

Monte Carlo simulation runs, a new solar power

scenario is generated, [3]. Next, by adding together

all of these iterations’ power numbers and guring

out their average, the predicted energy is found.

With regard to the predicted average power out-

put, this expected energy value is a useful indicator

for decision-makers because it takes into account

the inherent unpredictability in solar power gener-

ation. Mathematically, the expected energy (Ex-

pected) can be calculated as the mean of the power

values across all simulation iterations.

v) Constant k to convert UCF in Ps The Sys-

tem Factor, which is commonly expressed as (KW

hour / $), is the inverse of the energy cost. The

constant k should be adjusted based on the par-

ticular units and scale needed for the application

in question, [2]. In order to meaningfully com-

prehend the nancial consequences in a setting

of energy generation and consumption, the fac-

tor of conversion k basically lls the distinction

between the physical features of planned power

and the economic considerations in uncertainty

cost functions (UCF), [3]. The relation is termed

by Ps=k×E[UCF ]in which Psrepresents the

scheduled power, UCF represents the uncertainty

WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.11

Sergio Raul Rivera Rodriguez,

Ameena Al-Sumaiti, Tareefa S. Alsumaiti

E-ISSN: 2224-350X

Volume 19, 2024

Fig. 5: Expected energy from Monetecarlo scenar-

ios.

cost function, whereas kis the conversion constant.

Each scheduling pattern (A and B) will have a dif-

ferent behaviour in the dierente time instances.

In Figure 6 and Figure 7 are showed the kconstant,

where it is depicted the results of the analytical

calculation (red) and the montecarlos simulation

(green).

Fig. 6: Constant k to convert UCF in Ps (pattern

A).

vi) Uncertainity cost functions (analytical and

Montecarlo simulation) An essential tool for eval-

uating the nancial eects of uncertainties related

to variables in a system is the Uncertainty Cost

Function (UCF). While Monte Carlo simulations

oer a more in-depth examination of the stochastic

nature of uncertainties, the analytical model oers

a rapid insight into the overall functioning of the

system, [3]. The specic characteristics of the sys-

Fig. 7: Constant k to convert UCF in Ps (pattern

B).

tem itself, the uncertainties (such solar power uc-

tuations) and related economic factors will deter-

mine the details of the uncertainty cost functions.

Each scheduling pattern (A and B) will have a dif-

ferent behaviour in the dierente time instances.

In Figure 8 and Figure 9 are depicted the results

of the UCF analytical calculation (red) and the

UCF montecarlos simulation (green).

Fig. 8: Uncertianity cost function (Analytical and

Monetecario simulation).

4 Discussion and Conclusion

We acknowledge the necessity to update the vari-

ance and statistical factors regulating the analyti-

cal model in conjunction with our scheduled power

classication. The uctuation, a crucial factor in

determining the stability of the system, and re-

lated statistical measures need to be in line with

WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.11

Sergio Raul Rivera Rodriguez,

Ameena Al-Sumaiti, Tareefa S. Alsumaiti

E-ISSN: 2224-350X

Volume 19, 2024

Fig. 9: Uncertianity cost function (Analytical and

Monetecario simulation).

the changing face of power generation. These ana-

lytucal developments (Equations (7) through (29)

are revised in light of this realization, guarantee-

ing that the mathematical framework will con-

tinue to be accurate and exible enough to accom-

modate the dynamic conditions of contemporary

power systems.

In this way, unlike the traditional model, that

somewhat was considering the scheduled power

within the range of Pmin to Pmax, this expanded

approach, presented in this paper, categorizes the

Psinto three major regions. In summary, the three

regions will conduct the following formulation:

f1(Ps) =Cu(Pmax +Pmin

2−Ps)

f2(Ps) = Co

Pmax −Pmin P2

2−PsPmin −P2

min

2+

Pmax −Pmin P2

2−PsPmax −P2

max

2

f3(Ps) = Co(Ps−Pmax +Pmin

Keeping into consideration the likelihood of

distribution of the solar power that is accessible;

this computation produced the predicted value of

the electricity that the solar panels will inject. The

next stage was to use Monte Carlo simulations,

which are strong computational methods that al-

low the investigation of various possibilities by in-

troducing unpredictability and randomness.

It is dicult to optimize the distribution of en-

ergy in power networks that have a large amount

of renewable generation. The unpredictability of

renewable sources makes it dicult to achieve

the precise target functions that conventional

optimization techniques demand. Our research

presents a fresh analytical model that has been

veried by Monte Carlo simulations.

Acknowledgment:

This work is supported in part by ASPIRE under

the ASPIRE Virtual Research Institute

Program-Award Number VRI20-07.

References:

[1] B. -J. Yoon, X. Qian and E. R. Dougherty,

”Quantifying the Multi-Objective Cost of

Uncertainty,” in IEEE Access, vol. 9, pp.

80351-80359, 2021, doi:

10.1109/ACCESS.2021.3085486

[2] C. Tang, C. Liang and Y. Hu, ”A Reliable

Quadratic Programming Algorithm for

Convex Economic Dispatch with Renewable

Power Uncertainty,” 2018 8th International

Conference on Power and Energy Systems

(ICPES), Colombo, Sri Lanka, 2018, pp.

40-43, doi: 10.1109/ICPESYS.2018.8626913.

[3] J. Barco-Jiménez, G. Obando, H. R.

Chamorro, A. Pantoja, E. C. Bravo and J. A.

Aguado, ”In-Line Distributed Dispatch of

Active and Reactive Power Based on ADMM

and Consensus Considering Battery

Degradation in Microgrids,” in IEEE Access,

vol. 11, pp. 31479-31495, 2023, doi:

10.1109/ACCESS.2023.3248958.

[4] X. Yang, Y. Zhang, H. He, S. Ren and G.

Weng, ”Real-Time Demand Side Management

for a Microgrid Considering Uncertainties,” in

IEEE Transactions on Smart Grid, vol. 10,

no. 3, pp. 3401-3414, May 2019, doi:

10.1109/TSG.2018.2825388.

[5] S. Haberman, M. Z. Khorasanee, B. Ngwira

and I. D. Wright, ”Risk measurement and

management of dened benet pension

schemes: a stochastic approach,” in IMA

Journal of Management Mathematics, vol. 14,

no. 2, pp. 111-128, Apr. 2003, doi:

10.1093/imaman/14.2.111.

[6] P. Lu et al., ”Research on the Day-Ahead

Dispatch Strategy for Multi-Energy Power

Systems Considering Wind and PV

Uncertainty,” 2023 IEEE/IAS Industrial and

Commercial Power System Asia (ICPS Asia),

Chongqing, China, 2023, pp. 1280-1285, doi:

10.1109/ICPSAsia58343.2023.10294696.

[7] Q. Li, Q. Cao and Y. Zhang, ”Robust

Optimal Reactive Power Dispatch against

WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.11

Sergio Raul Rivera Rodriguez,

Ameena Al-Sumaiti, Tareefa S. Alsumaiti

E-ISSN: 2224-350X

Volume 19, 2024

Line Parameters Uncertainty,” 2018

International Conference on Power System

Technology (POWERCON), Guangzhou,

China, 2018, pp. 3729-3733, doi:

10.1109/POWERCON.2018.8602192.

[8] W. Lin and C. Zhao, ”Cost Functions Over

Feasible Power Transfer Regions of Virtual

Power Plants,” in IEEE Systems Journal, vol.

17, no. 2, pp. 2950-2960, June 2023, doi:

10.1109/JSYST.2022.3198278.

[9] B. -J. Yoon, X. Qian and E. R. Dougherty,

”Quantifying the Objective Cost of

Uncertainty in Complex Dynamical Systems,”

in IEEE Transactions on Signal Processing,

vol. 61, no. 9, pp. 2256-2266, May1, 2013, doi:

10.1109/TSP.2013.2251336.

[10] W. Zhu and Q. Wang, ”Distributed

Finite-Time Optimization of Multi-Agent

Systems With Time-Varying Cost Functions

Under Digraphs,” in IEEE Transactions on

Network Science and Engineering, vol. 11, no.

1, pp. 556-565, Jan.-Feb. 2024, doi:

10.1109/TNSE.2023.3301900.

[11] T Valencia, D Agudelo, D Arango, C Acosta,

S Rivera, D Rodriguez; A Fast Decomposition

Method to Solve a Security-Constrained

Optimal Power Flow (SCOPF) Problem

Through Constraint Handling; IEEE Access

99 (99), 36-60.

[12] J Bernal, J Neira, S Rivera; Mathematical

Uncertainty Cost Functions for Controllable

Photo-Voltaic Generators Considering

Uniform Distributions; WSEAS Transaction

on Mathematics 18 (1), 16-22.

Contribution of Individual Authors to the

Creation of a Scientic Article (Ghostwriting

Policy)

The authors equally contributed in the present re-

search, at all stages from the formulation of the

problem to the nal ndings and solution.

Sources of Funding for Research Presented in a

Scientic Article or Scientic Article Itself

This work is funded in part by ASPIRE

under the ASPIRE Virtual Research Insti-

tute Program-Award Number VRI20-07.

Conicts of Interest

The authors have no conicts of interest to

declare that are relevant to the content of this

article.

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WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.11

Sergio Raul Rivera Rodriguez,

Ameena Al-Sumaiti, Tareefa S. Alsumaiti

E-ISSN: 2224-350X

Volume 19, 2024