Optimal Sizing of Hybrid Generation Systems (Photovoltaic System and Energy Storage System) for Off-Grid Applications ^†

Abstract

This paper presents an optimal sizing strategy for a hybrid generation system combining photovoltaic (PV) and energy storage systems. To achieve this, the optimization problem is solved using the simplex method for linear programming, implemented through Python. The model considers test data on electrical energy demand and solar irradiation, alongside battery operating conditions such as state of charge (SOC) and upper and lower charge limits as key decision variables. Conventional PV system sizing serves as a benchmark to assess the effectiveness of the optimization, with particular attention given to the computational resources required for problem solving.The results obtained from the optimization method demonstrate a substantial improvement in the utilization of energy resources, both from the photovoltaic system and the energy storage system. This approach enabled the design of an optimized system based on the proposed model, which was further refined using Matlab/Simulink.

1. Introduction

Currently, the development of electrical generation systems based on renewable energies is of great importance in addressing the challenges posed by the reduction of CO₂ emissions and the search for sustainable and environmentally friendly alternative energies. Energy, along with resources such as water and transportation, among other factors, has a significant impact on development. These resources often become essential services in remote areas, thereby limiting the development of rural communities. The costs associated with conventional energy resources for remote areas, such as gas, coal, and other fossil-fuel-based resources, are generally much higher than in urbanized areas due to their challenging accessibility in most cases [1].

Considering the cost of electrical service, particularly for lighting, it can be more expensive in sparsely populated rural areas compared with those with more frequent access to the conventional electrical grid. In light of this issue, renewable energy generation systems have become a crucial resource within energy systems, offering a solution to environmental problems and reducing dependence on fossil resources. Additionally, an important aspect to emphasize is that they can be highly useful in the electrification of locations and areas that do not have easy access to energy services [2].

The issues associated with the use of renewable energy generation systems relate to the unavailability of resources 24 h a day, due to their stochastic nature and factors such as weather, which limit their use in isolated areas considering their inability to meet energy demand at all times. Considering these aspects, there arises a need to develop hybrid systems based on renewable energies, in which the power system operates with two or more energy resources, with at least one being a renewable energy source. Additionally, these systems can be complemented with conventional generation systems such as a diesel generator and/or energy storage systems, in order to supply energy demand during critical moments. This ensures the continuity and quality of service [3].

Considering the development of hybrid systems, various studies have been found in the relevant state of the art. Initially, hybrid systems were considered for early applications in telecommunications stations and economic activities. They were later implemented in various countries with a focus on rural electrification for areas lacking interconnection, where the extension of different distribution and transmission networks incurred high costs, as well as in distributed generation systems [4,5,6,7,8,9].

Applications such as water pumping systems using autonomous hybrid photovoltaic–wind systems in remote rural communities are proposed in [10]. A system enhancement is proposed in [11], which involves a hybrid photovoltaic–wind power system supplemented with a fuel cell, serving as a backup for a mobile dwelling. Based on this type of system, ref. [12] presented an adaptation of a battery system for this type of configuration, allowing for battery charging during periods of excess energy in the hybrid system, as well as execution of supplementary functions.

In the context of managing hybrid generation systems, various studies have been found in the literature, for example, in [13]. It encompassed an analysis related to the main benefits and motivations regarding the implementation of hybrid generation systems, considering the various opportunities associated with different renewable resources, in contrast with the different simulation and optimization tools available for the optimal management and development of diverse energy resources. To provide a broader perspective on the development of hybrid generation systems, the study in [14] presented a comprehensive review of the state of the art, considering aspects such as system architecture, energy storage systems, auxiliary generation components, software, algorithms, and economic and reliability criteria related to the optimization of these systems. Additionally, [15] offers an analysis of various structural types, addressing design opportunities and simulation based on optimization models for hybrid generation systems utilizing renewable energy sources.

As can be observed, hybrid energy systems have a wide range of applications in addition to their development potential. Given that these are electrical power generation systems, they may interact directly or indirectly with the traditional electrical grid. Consequently, two technology categories are identified: on-grid (grid-connected) and off-grid (standalone or non-grid connected) [16].

The simplex algorithm is a well-established and efficient method for solving linear programming problems, making it highly suitable for the optimal design of hybrid generation systems involving photovoltaic (PV) and energy storage components. These systems typically require the optimization of multiple decision variables, such as the sizing of PV panels and battery storage, while adhering to constraints related to energy demand, solar irradiance, and battery performance (e.g., state of charge limits). The simplex method is particularly adept at handling linear relationships, ensuring that the system is optimized for both performance and cost-effectiveness [17].

In the context of hybrid systems, where renewable energy generation and storage must be carefully balanced to meet demand and maximize resource efficiency, the simplex algorithm offers a robust solution by efficiently navigating feasible regions of the problem space to find an optimal solution [18].

Its computational efficiency is especially valuable in real-time decision-making or iterative processes, such as those required in dynamic environments with varying solar irradiance and energy demand. By employing the simplex algorithm, designers can ensure the system is both technically and economically optimized, a crucial factor for the widespread adoption of renewable energy technologies in modern power systems [19].

The optimal design of hybrid energy systems becomes a complex task due to the difficulty of accurately predicting the performance of these energy systems. This complexity arises for various reasons. Firstly, the optimization problem involves a large number of variables. Additionally, conflicting objectives are present, making the optimization problem complex, such as cost, performance, supply and demand management, network constraints, and other factors. Furthermore, another reason arises due to coupled nonlinearities, non-convexities, and mixed-type variables, which often eliminate the possibility of using conventional optimization methods to solve this type of problem [20].

2. Hybrid Electrical Power Generation Systems

Hybrid power generation systems can be defined as the combination of two or more energy conversion elements, such as electrical power generators or energy storage elements, or two or more fuels for the same element, which, when integrated, have the capability to overcome the limitations inherent to any of them [21].

The proposed definition of hybrid systems encompasses a wide range of possibilities, characterized by the integration of multiple energy conversion systems. This includes stationary energy systems, where at least one energy conversion element leverages renewable energy sources. A critical analysis of conventional energy systems, which are typically employed, is necessary to identify potential applications where a hybrid system could be effectively utilized as a substitute, thereby enhancing the overall efficiency and sustainability.

Based on their characteristics, hybrid electric power systems have a wide range of applications. The main applications include [20]:

• Remote area AC electrical grids, where traditional grid extension is not feasible.
• Integration of distributed energy resources in existing distribution infrastructure.
• Power supply solutions for isolated, rural, or special-purpose electrical loads.

In the context of hybrid generation systems, a typical scheme is shown in Figure 1. Here, the interaction between three different generation systems based on renewable energies is illustrated, using an energy storage system to supply a load.

3. Simplex Method Algorithm for Solving Linear Programming Optimization

This linear programming method provides a tool to systematically assess or examine the vertices of a feasible region, so in order to estimate and calculate the optimal value of the objective function [22]. The simplex optimization method is an iterative technique that geometrically corresponds to moving from one so-called feasible corner point to another, until the optimal feasible point is located. Slack variables are introduced to ensure that the corner points are feasible and remain within the solution region. Algebraically, the method involves transitioning from one feasible corner point to another by repeatedly identifying the pivot column, pivot row, and consequently, the pivot element within a sequence of matrix tableaux. Upon identifying the pivot element, a new tableau is created by pivoting (using the Gauss–Jordan method) around this element. This section considers the use of slack variables and pivoting within the context of the standard maximization problem [23]. The following characteristics are considered in the optimization model:

• The objective function is linear and maximized.
• The variables are non-negative.
• Structural constraints, all of the form:

  $$ax+by+...\le c,wherec\ge 0$$

  (1)

4. Methods

4.1. Hybrid System Modeling

The proposed methodology utilizes linear programming techniques to determine the optimal size of the photovoltaic generation system and energy storage system for an off-grid system, ensuring minimal costs and maximal efficiency. To achieve this, historical solar irradiance data and test energy consumption profiles will be utilized as inputs. The flowchart in Figure 2 provides a step-by-step illustration of the process to be developed, outlining the key stages and decision points involved in the optimization process.

Referring to the flowchart in Figure 2, the following processes are involved:

• Formulation of the optimization problem: the problem is defined, and in this practical scenario, a hybrid power generation system consisting of a photovoltaic system and an energy storage system is considered.
• Model parameterization and data acquisition: key parameters of the photovoltaic system and energy storage system models will be identified and collected, including solar irradiance data, energy storage capacity, and electrical load profiles.

The next steps of the proposed scheme are detailed in the following section.

4.2. Hybrid Power Generation System Design and Optimization Methodology

The electrical energy generated by the photovoltaic system ($P{V}_{size}$) exhibits a direct linear relationship with the system’s size, indicating that an increase in system size will result in a corresponding increase in energy production. For the purpose of designing the hybrid energy system, a total system efficiency of 20% is assumed, taking into account losses in the photovoltaic array, energy storage, and power conversion components.

The design of the hybrid energy system is based on the following key considerations:

• The design of the hybrid energy system must ensure that the sum of the photovoltaic generation ($P{V}_{gen}$) and battery discharge ($P_{discharge}$) is greater than or equal to the electrical load, guaranteeing a reliable and continuous power supply.
• In scenarios where the photovoltaic generation capacity exceeds the electrical energy demand, the excess energy can be effectively harnessed to charge the battery system ($P_{charge}$), thereby optimizing energy storage and reducing energy waste.
• If the excess photovoltaic generation exceeds the available capacity in the battery system, it can be assumed that the battery is fully charged, and it is necessary to reduce the photovoltaic generation. This defines the maximum amount of charge delivered.
• The amount of energy that can be discharged by the battery system must be less than the available capacity in the battery system.
• The state of charge of the battery system (SOC) is defined as $B_{state}\left(t\right)$, and the maximum charge is defined as $B_{max}$.
• The objective function is defined with the intention of minimizing the system cost, which is defined as the size of the photovoltaic system multiplied by the cost of the photovoltaic system, plus the cost of the battery system, which is related to the battery capacity and its cost, in addition to a penalty for the cumulative discharge of the battery system.

  $$mi{n}_{{PV-Bat}_{size}}(P{V}_{size}\times cos{t}_{PV}+B_{max}\times cos{t}_{bat}+penalt{y}_{value})$$

  (2)

For the solution of the linear programming model, Python programming tools have been utilized. In this context, the use of programming tools such as Python becomes highly valuable for addressing and solving optimization problems. Additionally, as an open-source software, it generally does not impose a significant computational burden in terms of performance.

5. Results and Discussion

The proposed scenario comprises a comprehensive dataset of 24 h solar irradiation and electrical energy demand profiles, carefully selected to facilitate in-depth analysis, testing, and validation of the proposed system.

The demand profile illustrated in Figure 3 represents a typical residential load pattern, characterized by a peak demand of approximately 1.4 kW at 21:00 p.m., and a minimum demand of around 0.2 kW, showing residential energy consumption characteristics, considering as a reference the load indexes used in the study carried out in [24].

The test solar irradiance profile depicted in Figure 4 showcases a representative diurnal cycle, with maximum irradiance levels attained during the daylight period, specifically between approximately 09:00 a.m. and 16:00 p.m., thereby illustrating the typical daily variation in solar radiation.

The analysis of the irradiance profile indicates that there are specific hours of the day when photovoltaic energy generation is not viable, requiring the utilization of energy storage systems to ensure a stable energy supply. Notably, during critical periods such as from 18:00 p.m. onwards, the absence of irradiance coincides with peak electrical energy demand, as depicted in Figure 3, highlighting the need for energy storage solutions to mitigate this mismatch.

By applying Equation (2), a comprehensive estimation of the photovoltaic system’s capacity is conducted, taking into account the monthly energy demand, to estimate the annual energy yield and ensure a reliable and efficient energy supply.

$$P_{PV}={\displaystyle \frac{Energ{y}_{monthly}\ast 12}{fp\ast 8760}}$$

(3)

By analyzing the demand profile presented in Figure 3 and applying a plant factor of 0.2, which is representative of photovoltaic systems, the estimated power rating of the photovoltaic system is determined to be 25.48 kW, ensuring a reliable and efficient energy supply.

For the battery system, a 0% State of Charge (SOC) has been set as the discharge limit, resulting in the following outcomes.

The operation of the hybrid generation system, as illustrated in Figure 5, shows a balanced interaction between energy generation and demand. During periods of maximum solar irradiance, the battery system charges, while during peak energy demand, the stored energy is discharged. This results in a reliable and resilient energy supply that meets the expected performance standards.

Table 1. Hybrid Generation System Sizing Results.

System	Value [kW]
PV system	15.1
Energy storage system	5.9

presents a comprehensive summary of the hybrid generation system sizing results, tailored to the specific conditions and requirements outlined in the analysis.

The results presented in Table 1 demonstrate that the optimized hybrid generation system design achieves a considerable reduction of approximately 40% in the initial photovoltaic system size, while also accurately determining the required battery system capacity. Furthermore, as depicted in Figure 5, the battery system exhibits the capability to reliably supply the demand during the specific time instants when it is required, thereby ensuring a robust and efficient energy supply.

To validate the obtained results, the optimization model developed in Simulink has been adapted from [25] (Figure 6). Using input data such as energy demand, solar resource availability, and a loss factor, the model estimates the design of the photovoltaic system. This process outputs key parameters, including the system’s capacity, the required number of solar panels, and the battery system capacity in ampere-hours (Ah).

The operating conditions considered include a 30 day period, 8 h of total daily solar irradiation, and a loss factor of 1.25. Additionally, for the battery system, a backup duration of 6 h is taken into account. The obtained results are detailed in

Table 2. Solar panel results.

Parameter	Value
Active Power of PV system	15.16 [kW]
Number of Solar panel	16
Ah of Battery per hour	336.8 [Ah]
Total battery capacity in Ah	2021 [Ah]

.

The results indicate that a photovoltaic (PV) system with 16 panels is required. For the energy storage system, the battery capacity must be at least 336.8 Ah.

6. Conclusions

An optimization model based on linear programming was developed using the simplex method. This approach does not demand significant computational resources and readily conforms to the characteristics of the mathematical model proposed for the hybrid generation system. Given the optimization model’s low computational demands, it has been employed both for estimating the energy production of the photovoltaic system relative to solar resource availability (solar irradiation) and for the design of the generation system. The latter includes considerations for the number of solar panels and the energy storage system’s capacity in ampere-hours (Ah). In conclusion, the adoption of this linear programming-based optimization model, particularly with the simplex method, demonstrates its efficacy in balancing computational efficiency with the accuracy of energy estimation and system design. This approach not only streamlines the modeling process, but also enhances the feasibility of implementing hybrid generation systems in scenarios where computational resources are limited, thereby promoting broader adoption of renewable energy solutions.