Uncertainty Cost Functions for Wave Energy

MUHAMMAD ATIQ UR REHMAN1, GINA IDARRAGA-OSPINA2, SERGIO RIVERA3

1Department of Electrical and Computer Engineering,

International Islamic University,

H-10, Islamabad,

PAKISTAN

2Facultad de Ingeniería Mecánica y Eléctrica,

Universidad Autónoma de Nuevo León,

MEXICO

3Departamento de Ingeniería Eléctrica,

Universidad Nacional de Colombia,

COLOMBIA

Abstract: - Wave energy is considered as an important energy source for people living near coastal areas to

meet their basic energy needs. Innovative and new technologies can be used for this energy utilization to run

the loads from sea waves. Due to its uncertain power availability, we can use uncertainty cost functions based

on probability distributions for the availability of the operation of the wave energy microgrid. These probability

distributions are based on several mapping models designed for the wave energy, wave height, and wave speed

mapping to make the energy more stable and reliable. In contrast to other renewable energy sources, wave

energy is inherently unpredictable, making it impossible to model with a single, globally applicable probability

distribution function (PDF). The prediction of the behavior of primary wave energy for this purpose is

completely based on several probability distribution functions (PDFs) which can be considered as the best for

all conditions. In this paper, we have used the Weibull-Rayleigh probability distribution model to develop the

uncertainty cost function for wave energy. The Monte-Carlo process is carried out to get the results supported

by the Rayleigh probability distribution model consisting of wave height, and uncertain cost histograms. Cost is

minimized by using the Weibull Rayleigh model for both overestimation and underestimation costs of wave

energy. Monte-Carlo simulation results are further compared with analytical calculation and error between

them.

Key-Words: - Microgrids, Renewable Energy Resources, Stochastic Processes, Wave Energy, Probability

Distribution, Variable Change Theorem, Weibull-Rayleigh distribution.

Received: May 6, 2023. Revised: May 8, 2024. Accepted: June 16, 2024. Published: July 30, 2024.

1 Introduction and State of Art

The energy which is obtained from sea waves'

movements and their motion is called wave energy

and this energy has many benefits in the context of

energy. The prediction of behavior of primary wave

energy for this purpose is completely based on

several probability distribution functions (PDFs)

which can be considered as the best of all cases. The

appropriate PDF from several choices based on

important factors with specific characteristics of the

given site, we can collect the wave energy with all

variable parameters, available wave energy

historical data, and analysis based on a modeling

approach to get the required outcomes, [1].

Using the kinetic and potential energy of ocean

waves, wave energy offers an enticing way to

guarantee a renewable energy source that is both

ecologically benign and sustainable for use

worldwide. Wave Energy Converters (WECs) are

carefully designed devices placed strategically in

oceans to convert wave motion into electrical

energy, [2], [3].

These WECs can be divided into different kinds

according to how they function, [4], [5], [6]:

 Point absorbers: These independent

constructions use internal machinery to

transform the mechanical energy they

absorb into electricity as they oscillate in

response to wave motion.

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 Overtopping devices: Featuring an inclined

ramp, the structure is overtopped by

oncoming waves, which fill a reservoir and

power turbines or other electricity-

generating devices.

 Oscillating water columns (OWCs): Air

inside partially submerged chambers is

compressed as a result of the chamber's

ability to absorb wave rise and fall. After

that, the compressed air powers turbines to

produce electricity.

But in order to precisely forecast wave behavior

and maximize WEC design and operation, complex

modeling approaches are required because of the

inherent diversity of wave parameters. Modeling

wave energy is essential for [7], [8]:

 Resource assessment: Calculating the

potential wave energy that is available at a

certain site.

 WEC design optimization: Creating a model

of the interaction between waves and

structures to help create reliable and

effective WECs.

 Forecasting power production: Estimating

how much electricity a WEC farm is likely

to produce.

Through the integration of many elements such

as bathymetry, wave climate, and WEC features,

these models facilitate the assessment of the

practicability and efficacy of wave energy initiatives

by scholars and practitioners. Nonetheless, effective

wave behavior prediction is essential for WEC

design and operation. In contrast to other renewable

energy sources, wave energy is inherently

unpredictable, making it impossible to model with a

single, globally applicable probability distribution

function (PDF).

This situation results from the interaction of

multiple factors [9], [10], [11]:

 Site-Specific Oceanography: Wave patterns

at a particular place are greatly influenced

by the water depth, bathymetry (seabed

composition), and the dominant wave

climate.

 Data Availability and Quality: The extent

and quality of past wave or wind speed

measurements play a major role in choosing

the right PDF.

 Goals for Modelling: The best PDF depends

on the model's goal, which could include

wave height prediction, energy capture

potential evaluation, or analysis of severe

wave events (such as rogue waves).

As such, choosing the best PDF requires a careful

assessment of these factors.

1.1 Common Probability Distributions in

Wave Energy Modelling

The most important several probability distributions

which can be used in order wave energy modelling

are as follows [12]:

Rayleigh Distribution: Wave amplitude is well-

modeled by this two-parameter function. The Rice-

Rayleigh distribution, which characterizes a

narrowband signal's envelope, serves as its basis.

When the underlying signal is completely

sinusoidal, the Rayleigh distribution becomes a

special instance and is especially useful when wind

speed data is provided. This is due to the fact that

wind speed and wave height are known to be

correlated.

Weibull Distribution: The three-parameter Weibull

distribution, which is comparable to the Rayleigh

distribution, is used to represent wave height or

wave energy as a function of wind speed. This

probability distribution has more degrees of freedom

as compared to the Rayleigh probability

distribution. In addition, it can apprehend wave

height fluctuations and a large range of wind speeds.

This probability distribution has characteristics

about its tail behavior and enables several cases of

modeling the systems having skewness of different

levels.

Generalized Pareto Distribution, or Pareto

Distribution: Pareto Distribution has the

characteristics of heavy-tailed along with a power

law tail. This characteristic of pareto probability

distribution supports to model severe wave events

for such enormous waves. This probability

distribution is more useful in calculating the energy

potential of waves. It has the capability to measure

and depict the probability of low-frequency and

high-impacted events of waves.

Lognormal Distribution: The lognormal probability

distribution function is used to model the wave

energy distribution and is more productive when the

data represents lognormal distribution and has

positively skewed a long tail. The natural positive

skewness due to the existence of massive wave

events can be frequently observed in wave energy

data. In this distribution, it is easier to capture

massive wave events than smaller ones.

Specialized Sea Wave Distribution: This

distribution function has the capability of capturing

the combined distribution of wave height and wave

period due to its additional characteristics of

capturing particular probability distributions. In

addition, it has the properties to demonstrate a more

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accurate image of the wave climate of a specific

place. So, they are producing frequent results in

order to show in-depth field observations or

numerical models.

It is a more critical procedure for data

availability and its quality which has a big impact

on probability distribution function (PDF) selection.

There are several statistical methodologies to select

the best distribution for a given dataset. For

example, moment matching, or maximum likelihood

estimation. In order to validate the selected model, a

thorough comparison with measured wave data is

used to guarantee the model’s correctness in

predicting wave energy behavior.

Considering more about the uncertainty cost

functions for wave energy, we have used different

PDFs in wave energy modeling in this research

work. Site specifications and site characteristics can

play an important role in deciding the right PDF in

the context of data availability and quality. We have

other statistical approaches for fitting the data in

order to determine a useful probability distribution

for data set availability. In the end, we need to

validate the model in order to find out its accuracy,

forecast ability, and wave energy speed behavior,

[3].

Based on the previous literature in this section I

and assumptions made in section II, we have

developed the analytical uncertainty cost functions

to deal with uncertainty behavior from wave energy

generation (section III).

2 Problem Formulation: Wave

Available Power from Wave

Energy

The Weibull and Rayleigh models for probability

distribution are more useful in order to do modeling

for the wave energy, wave height, and wave speed.

We can formulate a problem based on the data

available in the context of wave energy.

2.1 Probability of Sea Wave Height

Applying different probability distribution functions

(PDFs) is necessary to characterize the stochastic

nature of sea wave heights. The modified Rayleigh

distribution (sometimes called the Rice distribution)

is one such method. The amplitude of ocean waves

as a function of particular factors is effectively

modeled by this distribution.

The modified Rayleigh distribution's

mathematical basis is its capacity to depict a

narrowband signal's envelope. Essentially, it

expresses the amount that arises from the addition of

two separate parts, [13], [14] :

 Random component: Caused by wind seas,

swells, and local wind influences, among

other things, wave heights exhibit inherent

variability.

 Deterministic component: This component

shows foreseeable changes in wave heights,

which are frequently related to tidal effects

or the existence of an underlying current

that is always present.

Consequently, the modified Rayleigh

distribution is useful in modeling the combined

influence of these two contributing components in

the context of wave energy. It recognizes the

existence of a deterministic component that might

be impacted by tides or prevailing currents, on top

of a random component that is probably caused by

wind and other ephemeral causes.

However it is important to recognize that the

underlying assumptions determine whether the

modified Rayleigh distribution can be applied.

Among these presumptions are:

 Narrowband process: It is assumed that the

underlying wave field is narrowband, which

denotes a dominating wave frequency and a

minimal spectral width.

 Gaussian noise: It is assumed that the

random element has a Gaussian distribution,

[15].

The sea waves have different heights, speed,

and energy. They can be mapped based on the

probability distributions in order to model the wave

height, speed, and energy. We can modify one of

these probability distributions in order to represent

wave height, wave speed, and wave energy. For this

purpose, we can use the Rayleigh probability

distribution or the Rice probability distribution. We

can use these probability distributions in order to

model the wave energy amplitude which is the

function of some specific parameters related to

probability distribution, [16].

In this paper, we have used modified Rayleigh

distribution as a continuous probability distribution

function which describes the magnitude of signal

height. The signal height is basically the sum of the

two independent components of signals. One of

these components is a random signal having

Rayleigh probability distribution and the other

component is called a constant deterministic signal

component. Wave energy and wave height are the

combination of random components based on the

variability in natural waves and deterministic

component is based on tide or wind intensity, [5].

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The probability density function (PDF) of the

modified Rayleigh Distribution is given by:

(1)

where:

 is the PDF of the wave height h

 is the scale parameter that determines the

dispersion of the distribution

 is the displacement parameter that

represents the deterministic component

 Is the modified Bessel function of the

first type and order zero

In this probability distribution, we can use the

model that is mapped for the height of waves in the

ocean with some specific time period, and wave

location. The important factor that is to be

considered is ocean wave height distribution is

completely dependent on wave location and

meteorological conditions of this wave. In order to

formulate this into a model, we need observational

data or information. In this context, the formulated

model basically represents an accurate model of

wave energy with its specific and particular

location, [6].

Modeling sea wave elevation over a certain

temporal frame and geographic area is one use for

the modified Rayleigh distribution. It is crucial to

recognize that wave height distributions have

inherent spatiotemporal variability. This fluctuation

results from the interaction of multiple factors:

 Geographical location: The features of wave

fields at a particular site are greatly

influenced by bathymetry (seabed

topography), proximity to coastlines, and

local weather patterns.

 Meteorological conditions: Wave

production and propagation are dynamically

modulated by wind speed, wind direction,

and prevailing weather systems, resulting in

variations in wave height distribution.

Therefore, in order to accurately reflect the

wave energy supply at a given area, site-specific

data or validated numerical wave models are

essential.

Significant wave height (

) is used in this

study as the main indicator of wave energy. A

reliable indicator of the wave climate is Hs, which is

the average height of the highest one-third of the

waves in a given sea state. The variability of wave

height at a given location and time is statistically

characterized by the probability distribution

function (PDF) of

, commonly known as the sea

wave distribution.

It is important to acknowledge that there is not a

single PDF that can be used to all situations and

places that completely characterizes wave height

distribution. As was previously indicated, the

modified Rayleigh distribution is a useful tool, but

its usefulness is dependent on certain assumptions

that may not hold true in all situations.

The modified Rayleigh probability distribution

for this can be expressed in the following

probability density function (PDF).

(2)

where:

 is the PDF of the significant wave

height Hs

 is the scale parameter that determines the

dispersion of the distribution

We can describe this distribution as the

probability of observational data about the

significant wave height based on specific time

interval and location. This probability distribution is

considered as the function of and it represents the

variations in the wave heights. It is very important

to consider this modified Rayleigh probability

distribution as an approximation that has all types of

properties and characteristics of wave height

distribution, [17].

In practical cases and applications, we need to

develop and design more specific wave energy

models that are based on observational data and

location in order to optimize the model, [8]. We are

able to find the probability distribution of wave

height and wave power by using this distribution

model. The direction of the wind, depth of water,

seafloor variations, and topography are the

additional parameters including and affecting the

observational data and formulated models, [9], [10].

The modified Rayleigh probability distribution

is used to establish a probabilistic framework to

determine the probability of meeting a particular

significant wave height (Hs) in a given time frame.

A single parameter σ (sigma) is the foundation for

this purpose for mathematical measurement of the

variability or dispersion in wave heights. Critically,

we can recognize several inherent drawbacks in this

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approach, [13], [16]. A rough approximation is

observed in Rayleigh distribution, and it does not

represent the full range of properties in sea wave

height distribution in all situations. Alternative

models are available for more investigations when

there are deviations in basic assumptions about the

model like narrowband wave processes or Gaussian

noise.

Real-world applications of wave energy are

making the strategy to use observational data or

verified site wave models. Both data sources and

models must have the ability for other variables

inclusion have substantial impact on wave height

distribution, giving accessible potential wave

energy. Some extra factors influencing the wave

energy are as follows, [17], [18]:

 Wind direction: To determine wave

formation and propagation patterns, we

need to find the predominant wind direction

influenced by the distribution of Hs.

 Wate depth: The distribution of wave height

at a given site is affected by water depth and

seafloor topography wave properties which

are modulated by using bathymetry.

 Seafloor topography: The seabed

topography produces shoaling and

refraction of sea waves which influences

both the distribution of Hs and the extracted

amount of wave energy.

Wave energy supply and its variability at a

specific site is depicted more accurately by using

observational data and site-specific models

considering the above extra factors.

2.2 Relationship between Significant Wave

Height and Available Injection Power

It is well well-known, [19], [20], relationship

between wave height and power produced from it.

The power produced from sea wave energy is

injected into the power grid system as per energy

needed to fulfill the loads. The wave energy is

dependent on several factors like energy conversion

system design, site location and characteristics, and

sea atmosphere/climate. The basic relationship

between significant wave height and available

injection power can be formulated by describing the

effects of wave height on power generation.

In the context of wave energy conversion, the

relationship between significant wave height (Hs)

and the extractable electrical power that may be

added to the grid is complex and influenced by a

number of parameters as follows:

 Wave Energy Converter (WEC) Design:

The quantity of power that can be extracted

from a particular wave climate is greatly

impacted by the intrinsic design features of

the WEC, such as its power capture

efficiency and operational constraints.

 Site-Specific Characteristics: The wave

energy supply that is accessible at a given

area is mostly determined by bathymetry

(seabed topography), water depth, and local

wave climate.

 Sea Conditions: The potential wave energy

that is accessible is dynamically influenced

by prevailing weather patterns, which

include wind direction, speed, and wave

characteristics.

However, a simpler connection can be

developed to provide a basic grasp of the link

between wave height and extractable power. The

power generated from wave energy is found by

using the kinetic energy concept based on wave

motion. The wave height factor can influence

kinetic energy. The kinetic energy present in wave

motion is captured by wave energy. Hs is a crucial

sign of the wave energy potential in a given area. A

generalized approximation of the relationship

between Hs and the available wave power density

(P) is given by the following formula:

(3)

where:

 is the available Power

 is the conversion coefficient that takes

into account the efficiency of the capture

device energy and other factors related to

system design

 Is the density of seawater

 is the gravitational acceleration

 is the significant wave height, which is a

statistical measure of wave height

 is the wave period, which is the time

between two significant crests

 is the effective capture area of the device

This formula for power available is in its

simplified form but its exact formula or relationship

is dependent on specific factors of the energy

conversion system and sea waves site climate.

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Moreover, the efficiency of energy availability can

be affected by wave variations, mechanical effects,

and conversion system efficiency, [19].

In short, we can say that the height of the wave

is important in the context of finding the power

availability in wave energy. We can define the

relation in the form of the model by using this

formula in equation (3) based on system parameters

and other sea conditions, [20].

Although the formula that is provided provides

a first evaluation of how wave height affects

extractable power, it is critical to recognize that it

has inherent limitations. There is fluctuation in the

relationship between available wave power density

(Pw) and Hs based on a number of factors:

 Wave Energy Converter (WEC) Specifics:

The quantity of electrical power that can be

extracted from a certain wave climate is

greatly influenced by the design of the

power take-off (PTO) system and its

conversion efficiency.

 Site Conditions: The wave energy resource

that may be captured depends on elements

including bathymetry, ocean depth, and

dominant wave characteristics.

 Wave Variability: The efficiency of energy-

capture devices may be impacted by the

intrinsic stochastic nature of waves, which

are defined by fluctuations in height, period,

and direction.

Moreover, the maximum extractable power

under particular sea conditions is largely dependent

on the mechanical strength and operational

constraints of the WEC.

Significant wave height (Hs) is essentially a

crucial marker of the wave energy potential at a

specific site. A fundamental understanding of the

connection between Hs and available wave power

density is offered by the generalized formula. Real-

world applications, however, demand thorough

modeling techniques that take into account:

 Specifics of the WEC design, such as the

PTO system's effectiveness.

 Advanced wave modeling approaches to

account for extreme wave events and wave

variability;

 Site-specific features such bathymetry,

water depth, and local wave climate.

Researchers and engineers can create more

accurate evaluations of the wave energy resource

and the functionality of WEC systems under

different operating scenarios by taking these extra

factors into account, [21].

2.3 Uncertainty Cost of Wave Energy

In this research work, we have formulated an

analytical cost function which is derived from an

expression consisting of significant wave height,

probability distribution, and occurrence, [12]. The

formulated problem is solved by having an

analytical cost function which is obtained from

analytical expression derived by using the

significant height of sea waves and their probability

of occurrence.

There are the following two cases connected

with the uncertainty cost of wave energy.

To do this, the following expression must be

solved:

(4)

where:

 is the PDF of the of P

In this way, considering the following from

sections 2.1 and 2.2, we have:

↕

3 Problem Solution: Analytical

Development for Wave

Uncertainty Cost Functions

The problem presented in Section 2.3 can be solved

by using following mathematical analytical

development.

(6)

We can obtain the inverse of g and its derivative

as follows:

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(7)

= (8)

The changes in variable expression can be

applied as follows by using the Variable Change

Theorem:

Consider that as the probability density

function of x, and y.

Where y = g(x)

Then the probability density function of y can be

written as follows:

(9)

Based on variable change theorem expression,

we can calculate the following costs.

3.1 Penalty Cost Due to Underestimate

The underestimated expected cost can be written as:

Where:

(10)

Suppose that the tidal technology is injecting the

power with its maximum capacity, and this

maximum capacity value is range between 10-20

times more in value than the power which is

programmed or forecasted. Then the underestimated

expression for uncertainty cost in both cases are as

follows:

Case 1: 10 times more value

(11)

Case 2: 20 times more value

(12)

We can write it in its generic form as follows:

(13)

Where parameter n value can be found based on

experiments by using tidal energy technology.

3.2 Penalty Cost Due to Overestimate

The Overestimated expected cost can be written as:

Where:

(14)

4 Simulation and Validation

In Figure 1, the Monte-Carlo process is utilized in

order to get original simulation results supported by

Rayleigh probability distribution model based on

significant wave height, and uncertainty cost

histograms. Histograms are showing the

relationships between significant wave height, cost

due to underestimation and overestimation and

power. Cost is minimized by using the Weibull-

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Rayleigh model and then this model results are

compared with analytical results.

The simulation results show the costs associated

with estimated power in power electric vehicles

(PEVs) relevant to wave heights and dispatched

power [13]. We have compared the results obtained

from the Monte-Carlo simulation with the results

obtained from analytical development and

calculations.

Fig. 1: Weibull wind speed and Uncertainty cost

These simulation results show the expected cost

insights and the accuracy of these results by

comparing them with the analytical approach as

shown in Figure 1.

We have developed a total uncertainty cost

functions histogram based on both cases; the

overestimation and underestimation parts as shown

in Figure 2.

Financial cost of wind energy is much more

than the energy produced from any other resource

due to its highly uncertain behavior. We cannot

compare wind energy cost to other energy resources

cost due to low wind energy value. The Monte-

Carlo process is utilized in order to get original

simulation results supported by the Rayleigh

probability distribution model for significant wave

height, and uncertainty cost histograms as shown in

Figure 1. Histograms show the relationships

between significant wave height, cost due to

underestimation and overestimation, and wave

power. Significant wave height, wave power, cost

due to underestimation, and cost due to

overestimation are presented in Figure 1. In

overestimation of cost, the cost of wind energy is

estimated too high while its actual value is low. On

the other hand for underestimation of cost, the cost

of wind energy is estimated too low and its actual

value is high. Cost is minimized by using the

Weibull-Rayleigh model and the results are shown

in Figure 1.

Fig. 2: UCF histogram

The complete simulation process has several

steps in order to get the complete results of both

underestimated and overestimated uncertain costs.

These steps are basic steps for developing a Monte-

Carlo simulation in order to analyze power

estimation error cost for wave energy scenarios.

These steps are as follows:

a. Initialization of Constants:

The developed code in Matlab has an

initialization process of several constants like

dispatched power (PS), a factor of scaling, No.

of trials (N), costs for underestimation (CU),

costs for overestimation (CO), and K1

coefficient values.

b. Random Generation:

After initialization, the iteration loop

generates the significant wave height (Hs)

random values by using Rayleigh probability

distribution based on an Escala (a scale

parameter). The power (Pe) calculations are

done by using the parameters K1 and Hs. The

dispatched power Ps and generated power Pe

are then compared to get the costs for both

underestimation (CU) and overestimation

(CO).

c. Histogram Plots:

The developed code basically generates the

histogram plots by using significant wave

height, power, and both cost underestimation

and overestimation.

d. Total Cost Calculation:

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To get the total cost, we have added both the

costs, the cost for underestimation and the

cost for overestimation.

e. Analytical Calculation:

The analytically developed calculations are

used to estimate expected costs for both

underestimation and overestimation based on

the calculations of dispatched power Ps, all

coefficients, and a constant n. These results

and calculations are performed using an

outside iteration loop.

f. Output:

The output values are obtained by using the

command print in the Matlab code. This code

actually prints the mean values of costs for

both underestimation (xu1) and

overestimation (xo1) as follows:

1. Expected total cost based on Monte-Carlo

Simulation.

2. Expected total cost from Analytical

Calculation.

By using these two scenarios, we also calculated

the error between both the Monte-Carlo simulation

and Analytical calculations, [14].

In general, this Monte Carlo simulation looks

into the financial effects of inaccurate power

estimation in the context of wave energy. It provides

information about predicted expenses and the

accuracy of the analytical method in comparison to

the simulation by comparing the results gained

through simulation with those produced from an

analytical approach, [9], [10], [11].

This code simulation results show the costs

associated with estimated power in power electric

vehicles (PEVs) relevant to wave heights and

dispatched power. We have compared the results

obtained from the Monte-Carlo simulation with the

results obtained from analytical development and

calculations. These simulation results show the

expected cost insights and the accuracy of these

results by comparing them with analytical

methodology.

5 Conclusion

Wave energy is considered as a renewable energy

source which can be integrated with the grid. It can

provide the energy requirements for the population

near the coastal area. Innovative and new

technologies to capture wave energy are used for its

utilization for the loads to be run. Due to its

uncertain availability, we have used uncertainty cost

functions based on probability distributions. These

probability distributions are based on several

mapping models designed for wave energy, wave

height, and wave speed mapping. We have used the

Weibull-Rayleigh probability distribution model in

order to minimize the uncertainty cost of wave

energy and minimize the cost function based on

observational data or information. The formulated

model has represented an accurate model for wave

energy with its specific and particular location and

wave height. Simulation and analytical based results

generated an error which will be the future approach

to minimize it.

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DOI: 10.37394/232016.2024.19.24

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Gina Idarraga-Ospina, Sergio Rivera

E-ISSN: 2224-350X

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WSEAS TRANSACTIONS on POWER SYSTEMS

DOI: 10.37394/232016.2024.19.24

Muhammad Atiq Ur Rehman,

Gina Idarraga-Ospina, Sergio Rivera

E-ISSN: 2224-350X

275

Volume 19, 2024