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European Journal of Applied Sciences – Vol. 12, No. 3

Publication Date: June 25, 2024

DOI:10.14738/aivp.123.16920

Olusanya, O. O., & Olagunju, A. P. (2024). Mathematical Modeling for Optimizing Hydropower Production: A Review of

Mathematical Models, Operating Factors, And Case Studies. European Journal of Applied Sciences, Vol - 12(3). 18-28.

Services for Science and Education – United Kingdom

Mathematical Modeling for Optimizing Hydropower Production:

A Review of Mathematical Models, Operating Factors, And Case

Studies

Olamide O. Olusanya

ORCID: 0000-0001-6893-0472

Department of Computer Engineering,

Bells University of Technology, Ota, Ogun State

Ademola P. Olagunju

Department of Electrical/Electronics

and Telecommunication Engineering,

Bells University of Technology, Ota, Ogun State

ABSTRACT

Hydropower production plays a crucial role in the sustainable energy landscape,

and optimizing its operating factors is paramount for maximizing efficiency and

output. In this review study, we delve into the world of mathematical modeling to

explore the diverse approaches employed in enhancing hydropower production.

Five research questions are investigated including the mathematical model used,

operating factors, data types, time horizons, and case studies. In all, we analyze 22

primary studies to gain valuable insights into this vital field. Four key categories of

mathematical models were utilized in the studies: Physical-based Models,

Statistical Models, Optimization Models, and Artificial Intelligence (AI) Models.

Each of these models offers unique perspectives and techniques for optimizing

hydropower systems, reflecting the interdisciplinary nature of the field. The data

sources adopted exhibit interesting diversity, with a majority of 16 studies relying

on primary data, while two studies incorporate secondary data, and four studies

employ synthetic data. This broad range of data types contributes to the

robustness and accuracy of the findings. Investigating the geographical

distribution of case studies, intriguing insights are inferred. China emerges as the

dominant case study location, accounting for 55% of the studies. Nigeria follows

with 18.2%, while Spain, Korea, and Iran contribute 9% each, underscoring the

global significance of hydropower production optimization. Ranging from daily to

multi-year perspectives, the data time horizons employed in the review study

provide a comprehensive understanding of the dynamics and long-term trends in

hydropower systems.

Keywords: Hydropower, mathematical model, renewable energy

INTRODUCTION

As the world increasingly seeks sustainable and renewable energy sources, hydropower has

emerged as a key player in meeting energy demands (Gallego-Castillo and Victoria, 2021).

Hydropower production offers numerous advantages, including clean energy generation

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Olusanya, O. O., & Olagunju, A. P. (2024). Mathematical Modeling for Optimizing Hydropower Production: A Review of Mathematical Models,

Operating Factors, And Case Studies. European Journal of Applied Sciences, Vol - 12(3). 18-28.

URL: http://dx.doi.org/10.14738/aivp.123.16920

(Huang et al., 2022), storage capabilities (Chang et al., 2017), and flexibility in balancing

electricity grids (Huang et al., 2022). To ensure efficient and sustainable hydropower

production, optimizing the operating factors is of utmost importance. In this study, the realm

of mathematical modeling was delved into to explore the diverse approaches employed in

enhancing hydropower production. By investigating operating factors, data types, time

horizons, and case studies, we aim to provide valuable insights into this vital field.

Hydropower generation relies on a complex interplay of various factors, including water

availability, reservoir capacity, turbine efficiency, and environmental considerations (Bo et

al., 2017). Optimizing these factors can lead to increased energy production while minimizing

costs and environmental impacts. To tackle this optimization challenge, researchers have

employed mathematical models to analyze and optimize hydropower systems. In this context,

mathematical models provide a framework for understanding the complex dynamics of

hydropower generation and aid in decision-making processes. The objectives of this study are

threefold. First, identification and categorization of the mathematical models employed in

optimizing hydropower production. By understanding the various modeling approaches,

insights can be gained into their strengths and limitations. Secondly, investigation of the

operating factors was considered in these models, including water flow management,

reservoir operation, turbine control, and environmental factors. By examining the operating

factors, we can identify key parameters for enhancing hydropower production efficiency.

Finally, the data types and time horizons utilized in the studies were analyzed to explore the

temporal and spatial aspects of hydropower optimization.

The contribution of this study lies in consolidating and analyzing the existing literature on

optimizing operating factors for enhanced hydropower production. Literatures implementing

optimization measures for enhanced hydropower production were searched with necessary

search string and investigated. In all, 22 primary studies emerged from the search. By

synthesizing the findings, a comprehensive overview of the mathematical models employed,

the operating factors explored, the data types and time horizons utilized, and the geographical

distribution of case studies were provided. This study contributes to the knowledge base by

offering insights into the best practices and approaches for optimizing hydropower

production, fostering further research and innovation in this critical field. The remainder of

this paper is organized as follows. Section 2 provides an overview of the mathematical models

used in hydropower optimization. Section 3 delves into the search methodology. Section 4

discusses the result, while the research is concluded in section 5 with recommendations.

OVERVIEW OF MATHEMATICAL MODELS

A mathematical model is a representation or abstraction of a real-world system using

mathematical equations or relationships to describe its behavior or make predictions

(Gallego-Castillo and Victoria, 2021). By developing mathematical models, researchers can

simulate and analyze the performance of hydropower systems under different operating

conditions. Constraints, such as environmental considerations can also be incorporated into

the models. The models can then be used to identify the optimal values for the operating

factors that would maximize power generation while minimizing costs or environmental

impacts. These mathematical models may consider factors such as the water flow rate, head

(the height from which water falls), turbine efficiency, generator efficiency, and other relevant

parameters (Hanoon et al., 2023). The models may also take into account external factors,

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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 3, June-2024

such as weather patterns and water availability, to determine the most efficient operating

strategies. Mathematical models can be categorized into the following clusters:

Physical-based Models

These models are based on physical laws and principles governing the behavior of the

hydropower system components (Temam, 2001). For example, models can be developed to

simulate the fluid dynamics of water flow through turbines, considering factors such as

pressure, velocity, and turbulence. These models help optimize the design and operation of

turbines to maximize energy conversion efficiency. Bernoulli's equation is an example of

physical-based model that describes the conservation of energy along a streamline in a fluid

flow as

P +

1

2

ρu

2 + ρgh = Constant (1)

➢ where P is the pressure,

➢ ρ is the density of the fluid,

➢ u is the velocity,

➢ g is the acceleration due to gravity,

➢ h is the height above a reference point.

Statistical Models

Statistical models use historical data and statistical techniques to analyze and predict the

performance of hydropower systems (Huang et al., 2022). These models can identify patterns,

correlations, and trends in the data, enabling the identification of optimal operating

conditions and strategies. Statistical models can also be used for forecasting water

availability, which is crucial for efficient hydropower production planning e. g Linear

Regression:

Y = β0 + β1Χ1 + β2Χ2 + ⋯ + βnΧn (2)

➢ Y represents the performance variable (e.g., power output).

➢ X1, X2, ..., Xn represent the independent variables (e.g., water flow rate, turbine

efficiency, etc.).

➢ β0, β1, β2, ..., βn are the coefficients that need to be estimated.

Optimization Models

Optimization models involve formulating the problem of maximizing hydropower production

as an optimization problem (Lu et al., 2021). Mathematical techniques such as linear

programming, nonlinear programming, or dynamic programming can be applied to find the

optimal values for operating factors. These models consider various constraints, such as

turbine capacity, environmental regulations, and energy demand, to identify the operating

strategies that maximize power output.