A Day-Ahead Optimal Battery Scheduling Considering the Grid Stability of Distribution Feeders

Abstract

This study presents a comprehensive framework for optimizing energy management systems by integrating advanced methodologies for weather forecasting, energy cost analysis, and grid stability using a mixed-integer linear programming (MILP) algorithm. A novel approach is proposed for day-ahead weather forecasting, leveraging real-time data extraction from reliable weather websites and applying clear sky modeling to estimate photovoltaic (PV) generation with high accuracy. By automating weather data acquisition, the methodology bridges the gap between weather predictions and practical energy management, providing utilities with a reliable tool for operating and integrating renewable energy. The optimization framework focuses on minimizing the utility bill by analyzing a distribution feeder representative of Australia’s energy infrastructure, incorporating time-of-use (TOU) and flat tariff systems across eight Australian states to simulate realistic energy costs. Furthermore, voltage constraints are applied within the optimization framework to maintain system stability and improve voltage profiles, ensuring both technical reliability and economic efficiency. The proposed framework delivers actionable insights for utility industries, enhancing the scheduling of battery energy storage systems (BESS) and facilitating the integration of renewable energy into the grid.

1. Introduction

Managing grid stability and ensuring reliable grid operation is a crucial challenge for the utility industry due to the increasing amount of renewable energy sources (RES) [1]. Moreover, a surge in electricity rates has become unavoidable due to the increase in energy prices as the consumption in residential building is increasing [2]. Battery energy storage systems (BESS) offer an effective solution for addressing grid volatility and intermittency by storing surplus energy from the grid and discharging it when it is needed [3]. Among all the renewable energy sources, a photovoltaic (PV) system is the most commonly recognized form of distributed generation (DG). It is essential to integrate a BESS alongside it [4]. Intermittency of PV generation is quite inflexible due to climate’s nature, which create uncertainties in grid operation [5,6]. Other uncertainties involve load profiles, electricity prices, and so on [7]. The effective scheduling of BESS with PV systems is a critical challenge to address these uncertainties.

The optimization challenges associated with BESS have grown more complex due to issues such as discontinuity, non-differentiability, highly nonlinear objective functions, and constraints. Consequently, there has been a growing interest in applying metaheuristic bases optimization algorithms such as genetic algorithm [8,9], particle swarm optimization (PSO) [10,11], and simulated annealing (SA) [12] to address these complexities. In parallel with metaheuristic approaches, machine learning (ML) techniques, particularly reinforcement learning (RL) and deep reinforcement learning (DRL), have emerged as promising tools to optimize BESS-scheduling strategies. These approaches focus on maximizing rewards and guiding agents through states and actions in dynamic, uncertain environments. The corresponding objective functions or reward formulations often aim to maximize BESS reliability [13], profit [14], operational profit [15], and minimize electricity costs [16], among other goals. In this context, the Markov Decision Process (MDP) [17] has emerged as a widely used method for modeling the sequential decision-making process of battery charging and discharging over a defined period. This process is typically addressed using advanced RL techniques, such as Twin Delayed Deep Deterministic Policy Gradient (TD3) [18], Proximal Policy Optimization (PPO) [19], Q-learning [15], policy-based algorithms [20], and Deep Q-Network (DQN) [21]. Linear Programming (LP) has widely been employed to determine the optimal charging and discharging schedules [22,23,24], as well as to identify the optimal solutions for electricity pricing, feed-in tariffs in order to minimize overall costs [25]. Moreover, mixed-integer linear programming (MILP) has been used to solve the BESS-scheduling problem considering the PV curtailment and load shedding [26]. An MILP-based bi-level optimization scheduling is proposed considering the wind farm uncertainties to reduce daily electricity cost and emission [27]. A day-ahead charging and discharging strategy based on MILP is proposed to optimize the energy cost where an artificial neural network (ANN)-based method is used to forecast the renewable energy output based on historical data [28]. Another area of interest is the application of machine learning for short-term load forecasting. Approaches like Federated Model-Agnostic Meta Learning (Meta-learning) have shown promise in forecasting load in distribution transformer supply zones, where energy demand fluctuates dynamically [29]. This technique allows for more accurate forecasting by leveraging data from multiple sources, while also ensuring privacy and adaptability to changing energy conditions. These forecasting methods are essential for improving the accuracy of BESS scheduling, as they provide better predictions of energy supply and demand, allowing for more efficient battery operation. In recent years, deep learning-based models and their advanced variants, including Long Short-Term Memory (LSTM) [30], Bi-directional Long Short-Term Memory (Bi-LSTM) [31], Two-Dimensional Convolutional Neural Networks (2D-CNNs) [32], and hybrid models combining Convolutional Neural Networks (CNNs) with LSTM (CNN-LSTM) [33], have gained significant popularity among researchers. These models are favored for their exceptional computational efficiency, strong ability to fit nonlinear relationships, resilience to abnormal data, and robust parallel computing capabilities. However, deep network-based methods generally require a large amount of training data, and insufficient data can lead to overfitting problems that may undermine the accuracy of predictions [34]. Unfortunately, end users in areas served by distribution transformers, particularly those with newly installed smart meters, often do not have enough historical load data. Mixed-integer nonlinear programming (MINLP) is proposed to solve a tri-objective problem considering uncertainties [35]. These methods have gained traction in addressing challenges related to renewable generation forecasting, often leveraging advanced machine learning algorithms for more accurate predictions. All these studies consider historical data for renewable generation forecasting. Although different methods have been proposed to solve multi-objective functions, voltage profile improvement within set boundaries remains a critical challenge while performing BESS scheduling. Therefore, the major contributions of this article are given below:

• A novel approach for forecasting day-ahead weather conditions is introduced, utilizing real-time data directly from reliable weather forecasting websites, eliminating the need for complex data processing.
• This real-time integration ensures accurate and up-to-date weather information, essential for energy management systems relying on renewable sources like solar PV.
• The clear sky model is applied to convert weather data into precise estimates of potential PV generation, accounting for geographical and meteorological conditions.
• The study models a distribution feeder representative of Australia’s energy infrastructure, incorporating real-world economic and regulatory frameworks like the TOU tariff system and flat tariff system across eight Australian states.
• Tariff structures are considered to simulate realistic energy costs, offering insights into how electricity prices impact energy storage scheduling and usage.
• Voltage constraints are included in the optimization framework to ensure system stability, preventing fluctuations that could destabilize the grid or damage equipment. By considering voltage constraints, the optimization not only focuses on economic benefits but also ensures the technical stability and reliability of the system.

2. Solar Generation Forecasting

Web scraping involves automatically extracting data from websites. For weather forecasting, scraping provides a way to gather real-time weather data from public application programming interface (APIs) or web pages, often offering insights into current conditions, hourly forecasts, and long-term predictions. While APIs are often the preferred method for obtaining weather data, many websites also present detailed forecast information through HTML tables, making them ideal candidates for scraping. This approach can be useful for applications where an API may be unavailable, or when a developer needs specific, localized data that is displayed in a particular format on a website. Selenium is a powerful web automation tool used for testing and scraping dynamic web content. Unlike traditional scraping techniques that work with static HTML, Selenium interacts with websites in a way that mimics real user behavior. Selenium is used to extract hourly weather forecast data for Brisbane, Australia, from Wunderground [36].

Workflow for Weather Data Extraction:

• The first step in the process is determining the date for the forecast. The script calculates the next 24 h using Python’s datetime module. This allows the system to dynamically generate a URL that corresponds to the desired forecast date.
• Automate the web-browsing process, Selenium WebDriver is configured to use the Firefox browser in headless mode. Headless mode means that the browser operates without opening a visible user interface, which helps improve performance, particularly when running automated tasks in the background.
• The script generates the correct URL for Wunderground’s hourly forecast page by embedding the calculated date into the URL structure. Once the browser navigates to the page, a short delay is introduced to ensure that the content has fully loaded before any data extraction begins.
• The script identifies the HTML table containing the forecast information. Using XPath, Selenium locates the table and begins extracting data. It first captures the table headers, which represent the forecast attributes (such as time, temperature, and humidity). Then, the individual rows of the table, which contain the actual forecast data for each hour, are processed. The extracted data table is shown in

  Table 1. Extracted weather data from Wunderground (16 October 2024).

  Time	Conditions	Temp.	Feels Like	Precip	Amount	Cloud Cover	Dew Point	Humidity	Wind	Pressure
  12:00 am	Clear	75 °F	79 °F	6%	0 in	15%	70 °F	85%	3 mph E	29.90 in
  1:00 am	Mostly Clear	75 °F	79 °F	6%	0 in	26%	70 °F	86%	2 mph ENE	29.88 in
  2:00 am	Mostly Clear	74 °F	76 °F	7%	0 in	28%	70 °F	87%	2 mph ENE	29.87 in
  3:00 am	Mostly Clear	73 °F	75 °F	7%	0 in	24%	70 °F	90%	2 mph E	29.86 in
  4:00 am	Clear	72 °F	75 °F	8%	0 in	17%	70 °F	92%	1 mph SW	29.87 in
  5:00 am	Clear	72 °F	74 °F	8%	0 in	18%	70 °F	92%	1 mph WSW	29.88 in
  6:00 am	Sunny	74 °F	77 °F	6%	0 in	18%	70 °F	88%	1 mph SSW	29.90 in
  7:00 am	Sunny	78 °F	83 °F	4%	0 in	9%	71 °F	79%	2 mph SSW	29.91 in
  8:00 am	Sunny	82 °F	87 °F	1%	0 in	16%	71 °F	69%	2 mph E	29.91 in
  9:00 am	Sunny	85 °F	90 °F	0%	0 in	12%	70 °F	60%	2 mph N	29.90 in
  10:00 am	Sunny	88 °F	94 °F	0%	0 in	8%	69 °F	53%	3 mph NNE	29.88 in
  11:00 am	Sunny	91 °F	97 °F	0%	0 in	5%	68 °F	47%	4 mph NE	29.86 in
  12:00 pm	Sunny	94 °F	100 °F	0%	0 in	2%	68 °F	43%	5 mph ENE	29.84 in
  1:00 pm	Sunny	95 °F	101 °F	0%	0 in	4%	67 °F	40%	6 mph ENE	29.81 in
  2:00 pm	Sunny	96 °F	102 °F	0%	0 in	10%	66 °F	38%	8 mph E	29.80 in
  3:00 pm	Sunny	96 °F	101 °F	0%	0 in	11%	66 °F	38%	9 mph ENE	29.78 in
  4:00 pm	Sunny	94 °F	99 °F	0%	0 in	10%	66 °F	41%	11 mph NE	29.77 in
  5:00 pm	Sunny	91 °F	96 °F	0%	0 in	4%	67 °F	46%	11 mph NE	29.78 in
  6:00 pm	Sunny	87 °F	92 °F	0%	0 in	3%	68 °F	53%	10 mph NE	29.80 in
  7:00 pm	Clear	84 °F	89 °F	1%	0 in	4%	69 °F	60%	8 mph NE	29.83 in
  8:00 pm	Clear	82 °F	87 °F	2%	0 in	10%	70 °F	67%	6 mph NE	29.85 in
  9:00 pm	Clear	81 °F	86 °F	3%	0 in	12%	70 °F	71%	5 mph NNE	29.86 in
  10:00 pm	Clear	80 °F	84 °F	4%	0 in	16%	71 °F	74%	4 mph NNE	29.87 in
  11:00 pm	Clear	78 °F	83 °F	5%	0 in	15%	71 °F	77%	3 mph NNE	29.87 in

  .

From the weather-forecasting table, cloud coverage ($CC$) is extracted to calculate the solar radiation. The core of the solar radiation calculation relies on a clear sky model, which determines the theoretical maximum solar radiation. The solar radiation is the sum of the beam radiation and the diffused radiation [37]. Therefore, the first step in developing the model was to define the equation for horizontal beam radiation, which is shown below:

$$G_{cb}=1339{\tau }_b\cos {\theta }_z$$

(1)

where ${\tau }_b=$ atmospheric transmittance beam radiation; ${\theta }_z$ = angle of incidence

The diffusion radiation is defined in the equation below:

$$G_{cd}=1339{\tau }_d\cos {\theta }_z$$

(2)

where ${\tau }_d=$ transmission co-efficient for diffuse radiation

Atmospheric transmittance beam radiation is defined in the equation below:

$${\tau }_b=a_o+a_1{e}^{-k/\cos {\theta }_z}$$

(3)

Transmission coefficients for diffuse radiation for clear days are defined in the equation below:

$${\tau }_d=0.271-0.294{\tau }_b$$

(4)

The constants $a_o$, $a_1$, and k are not arbitrary but are derived from empirical equations based on experimental data and the specific atmospheric conditions for different altitudes [37]. The correction factors $r_o$, $r_1$, and $r_k$ are introduced to account for these environmental variations. These correction factors are defined in Equations (5)–(7), respectively. These correction factors ensure that the model is adaptable and provides accurate solar radiation predictions across a variety of environmental conditions. They also help align the model’s predictions with real-world observations, improving the accuracy of solar energy generation estimates. A is altitude of the observer in km.

$$a_o=(0.4237-0.00821{(6-A)}^2) r_o$$

(5)

$$a_1=(0.4237+0.00595{(6.5-A)}^2) r_1$$

(6)

$$k=(0.2711+0.01858{(2.5-A)}^2) r_k$$

(7)

Incidence angle of horizontal surfaces is defined in the equation below:

$$\cos {\theta }_z=\cos \phi \cos \delta \cos \omega +\sin \phi \sin \delta$$

(8)

where $\phi$ = Latitude; $\omega$ = Hour angle; $\delta$ = Declination angle

To determine the declination angle, the equation below is used, where dd is the day of the year.

$$\delta =23.45\sin \left(360{\displaystyle \frac{284+dd}{365}}\right)$$

(9)

The hour angle represents the sun’s apparent movement across the sky throughout the day measured in degrees.

The solar radiation is the sum of the two as shown below:

$$G_0=1339{\tau }_b\cos {\theta }_z+1339{\tau }_d\cos {\theta }_z$$

(10)

The total solar radiation can be calculated as follows:

$$G_{total}=G_0(1-CC)$$

(11)

where $CC$ represents the Cloud Cover column of Table 1.

3. Mathematical Modelling

3.1. Microgrid Modelling

Figure 1 illustrates a schematic diagram of a renewable energy community that integrates both prosumers and consumers into a shared low-voltage distribution network. The community is connected to a larger distribution network through a medium-voltage (MV) transformer. The low-voltage distribution network carries power from the MV transformer and distributes it to various entities within the renewable energy community. It acts as the main line that branches out to supply individual loads, including prosumers and consumers. This community-based microgrid is designed to operate sustainably by utilizing local renewable energy sources, energy storage, and energy-sharing capabilities among participants. The community includes prosumers—members who both consume and generate electricity—and consumers, who only consume power. The variables and parameters are discussed briefly in the following section.

3.2. Objective Function

The objective function is to minimize the electricity bill $E_{bill}$ within a 24 h timeframe, which is the difference between costs and revenue. If $E_{bill}$ is positive, then the amount is to be paid to the utility company. If $E_{bill}$ is negative, then the amount is received in terms of energy credits.

$$E_{bill}\left(x\right)=\sum _{t=1}^T\left(P_{import}{c}_{import}\right(t)+P_{chr}{c}_{chr}\left(t\right)+P_{dich}{c}_{dich}\left(t\right))-\sum _{t=1}^T{P}_{export}{c}_{export}\left(t\right)$$

(12)

where $x$ is a vector of decision variables; $P_{import}$ is the power imported from the main grid, and $c_{import}$ is the cost of imported energy; $P_{export}$ is the exported power to the main grid, and $c_{export}$ is the cost of exported energy; $P_{chr}$ is the battery-charging power, and $c_{chr}$ is the cost per kWh for charging the battery; $P_{dich}$ is the battery-discharging power, and $c_{dich}$ is the cost per kWh for discharging the battery.

3.3. Set of Constraints

3.3.1. Power Exchange and Balance Constraint

To schedule the battery energy storage system in a microgrid, a set of constraints are required related to networks physical operation.

The grid cannot import and export power simultaneously.

$$0\le {\alpha }_{import}+{\alpha }_{export}\le 1$$

(13)

${\alpha }_{import}$ and ${\alpha }_{export}$ are binary state variables that control whether the system is in import or export mode.

The main grid can only import and export power up to the maximum import ($P_{import}^{max}$) and export power ($P_{export}^{max}$) defined by the utility.

$$0\le P_{import}\left(t\right)\le {\alpha }_{import}\left(t\right)P_{import}^{max}$$

(14)

$$0\le P_{export}\left(t\right)\le {\alpha }_{export}\left(t\right)P_{export}^{max}$$

(15)

To satisfy the power balance, the total power generation must be equal to the total power demand.

$$P_{import}\left(t\right)+P_{dich}\left(t\right)+P_{pv}^{fcst}\left(t\right)=P_{chr}\left(t\right)+P_{export}\left(t\right)+P_{load}^{fcst}\left(t\right)$$

(16)

where $P_{chr}$ and $P_{dich}$ are the battery-charging and -discharging power; $P_{pv}^{fcst}$ and $P_{load}^{fcst}$ are the forecasted PV and load power.

3.3.2. Voltage Constraints

To maintain the grid stability, the voltage at each bus needs to remain within the allowable limits set by the utility industry.

$$V_{min}\le V\left(t\right)\le V_{max}$$

(17)

where $V_{min}$ and $V_{max}$ are the minimum and maximum voltage limits.

3.4. Battery Energy Storage System

The overall efficiency of the battery energy storage system (${\eta }_{bess}$) is the product of efficiencies of three core components: coupling transformer, converter and the battery pack.

$${\eta }_{bess}={\eta }_{tr}{\eta }_{con}{\eta }_{bat}$$

(18)

where ${\eta }_{tr}$ is the efficiency of the coupling transformer, which is used to step up or step down the voltage levels between the battery and the grid; ${\eta }_{con}$ represents the efficiency of the converter system, which converts the AC output of the grid to DC input into the battery during charging and vice versa with discharging; ${\eta }_{bat}$ is the efficiency of the battery which represents how the battery can store or release energy efficiently.

The battery efficiency (${\eta }_{bat}$) can be expressed as the square root of the round-trip efficiency (${\eta }_{rt}$) as follows:

$${\eta }_{bat}=\surd {\eta }_{rt}$$

(19)

The round-trip efficiency is the ratio of the energy retrieved from the battery to the energy going into it. It is a common practice to have charging and discharging efficiency to be the same. Different battery technologies have different round-trip efficiencies: lead acid batteries have 92%, lithium-ion batteries have 94%, and LiFePO4 batteries have higher efficiency of 97% [17]. Again, the typical transformer and converter efficiencies are greater than 97%.

State of charge (SOC) refers to the amount of energy stored in the battery compared to its maximum available energy capacity ($E_v$). The SOC can be updated recursively based on the power input during charging and the power output during discharging.

$$SOC\left(t\right)={\displaystyle \frac{{\eta }_{BESS}{P}_{chr}\left(t\right)}{E_v}}-{\displaystyle \frac{P_{dich}\left(t\right)}{{\eta }_{BESS}{E}_v}}+{SOC}_{in} for t=1$$

(20)

$$SOC\left(t\right)={\displaystyle \frac{{\eta }_{BESS}{P}_{chr}\left(t\right)}{E_v}}-{\displaystyle \frac{P_{dich}\left(t\right)}{{\eta }_{BESS}{E}_v}}+{SOC}_{t-1} for t>1$$

(21)

where ${SOC}_{in}$ refers to the initial state of charge.

For optimization purposes, the SOC should be limited to predefined range.

$${SOC}_{min}\le SOC\left(t\right)\le {SOC}_{max}$$

(22)

where ${SOC}_{min}$ and ${SOC}_{max}$ are the minimum and maximum allowable state of charge, respectively.

For a balanced system, the final value of SOC at the end of period T should be the same as the initial value.

$$SOC\left(t=T\right)={SOC}_{in}$$

(23)

The charging and discharging operations cannot happen simultaneously.

$$0\le {\alpha }_{chr}\left(t\right)+{\alpha }_{dich}\left(t\right)\le 1$$

(24)

where ${\alpha }_{chr}$ and ${\alpha }_{dich}$ are the state variables for charging and discharging operations.

Equations (25) and (26) impose limits on the charging and discharging power of the battery energy storage system (BESS), ensuring that they do not exceed the rated BESSs power ($P_r^{BESS}$).

$$0\le P_{chr}\left(t\right)\le {\alpha }_{chr}\left(t\right)P_r^{BESS}$$

(25)

$$0\le P_{dich}\left(t\right)\le {\alpha }_{dich}\left(t\right)P_r^{BESS}$$

(26)

3.5. Matrix Formulation

When formulating an optimization problem for a mixed-integer linear programming (MILP) solver, the mathematical problem is typically expressed in matrix form. The problem is structured as follows:

$$\begin{gathered} \underset{n}{\min }{f}^{T}x \\ subject to \left\{\begin{array}{c}Ax\le b\\ A_{eq}x=b_{eq}\\ l_b\le x\le u_b\end{array}\right. \end{gathered}$$

(27)

where $f$ is a vector of co-efficient representing the cost and is a vector of solution variables, and $f^{T}x$ represents a linear combination of the decision variables that defines the objective; $A$ is a matrix representing the co-efficient of the inequality constraints, and $b$ is a vector representing the upper bounds for each inequality constraint; $A_{eq}$ is a matrix representing the co-efficient of the equality constraints, and $b_{eq}$ is a vector representing the upper bounds for each equality constraint; $l_b$ and $u_b$ are the lower and upper bound for each decision variable.

The $f$ and$x$ can be represented as follows:

$$f=[\left\{c_{chr}\left(t\right)\right\};\left\{c_{dich}\left(t\right)\right\};\left\{c_{import}\left(t\right)\right\};\{-c_{export}\left(t\right)\left\}\right]$$

(28)

$$x=[\left\{P_{chr}\left(t\right)\right\};\left\{P_{dich}\left(t\right)\right\};\left\{P_{import}\left(t\right)\right\};\{P_{export}\left(t\right)\left\}\right]$$

(29)

The optimization program runs for 24 h, which includes four decision variables. Therefore, a total of 96 decision variables has to be optimized in each cycle.

3.6. Cost Estimation

The annualized life cycle cost of energy storage (LCCOES) gives a total cost of a storage in a yearly figure for the whole project life span. This cost involves capital investment and operations and maintenance cost and can be expressed as follows [38]:

$$LCCOES=C_{inv}+C_{OM}$$

(30)

where $C_{inv}$ is the total investment cost, and $C_{OM}$ is the total operation and maintenance cost.

The total investment cost can be expressed as follows:

$$C_{inv}=C_{invest,in}\ast RF$$

(31)

where $C_{invest,in}$ is the initial investment cost, and $RF$ is the recovery factor.

The recovery factor can be expressed as follows [39]:

$$RF={\displaystyle \frac{i{(1+i)}^y}{{(1+i)}^y-1}}$$

(32)

where $i$ is the discount rate, and $y$ is the service life.

The initial investment cost consists of the cost of the associated devices, such as storage container ($C_{sc}$), power conversion devices ($C_{pc}$), and transformer and protection devices ($C_{pd}$).

$$C_{invest,in}=C_{sc}+C_{pc}+C_{pd}$$

(33)

The cost of storage container can be represented by Equation (34):

$$C_{sc}=c_s\left({\displaystyle \frac{E_r}{{\eta }_{rt}\ast {DOD}_{max}}}\right)$$

(34)

where $c_s$ refers to the cost of storage container in AUD/kWh, $E_r$ is the rated capacity in kWh, and ${DOD}_{max}$ is the maximum depth of discharge.

The cost of the power converter is represented by Equation (35):

$$C_{pc}=c_{pc}\ast P_r$$

(35)

where $c_{pc}$ refers to the cost of power converter in AUD/kW

The cost of protection and other remaining devices is expressed in Equation (36):

$$C_{pd}=c_{pd}\ast P_r$$

(36)

$c_{pd}$ refers to the cost of the protection devices in AUD/kW.

The operation and maintenance costs are divided into two categories: a fixed cost ($C_{fixed}$), which remains constant throughout the lifetime of the project regardless of the operational phase of the energy storage (ES) technology, and a variable cost ($C_{variable}$), which fluctuates depending on the operational phase of the ES technology.

The fixed operation and maintenance cost can be expressed in Equation (37):

$$C_{fixed}=c_{fixed}\ast P_r\ast RF\ast \left(\sum _{m=1}^{m=y}{\displaystyle \frac{{(1+V_{om})}^m}{{(1+i)}^m}}\right)$$

(37)

where $c_{fixed}$ refers to the fixed operation cost expressed in AUD/kW; $V_{om}$ refers to the annualized factor rate.

The annual variable operation and maintenance cost is represented in Equation (38):

$$C_{variable}=B_{elec}\ast E_r\ast {\displaystyle \frac{d}{24}}\ast {\displaystyle \frac{h_d}{{\eta }_{rt}}}\ast RF\ast \left(\sum _{m=1}^{m=y}{\displaystyle \frac{{(1+V_{elec})}^m}{{(1+i)}^m}}\right)$$

(38)

where $B_{elec}$ refers to the electricity price in AUD/kWh; $d$ is the total working days per year; $h_d$ is the number of working hours per day; $V_{elec}$ refers to annual change rate $B_{elec}$.

Then, the total operation and maintenance cost can be expressed as follows:

$$C_{OM}=C_{fixed}+C_{variable}$$

(39)

The annualized energy output from a storage over the lifetime of the system can be expressed as in Equation (40):

$$E_{out}=E_r\ast {DOD}_{max}\ast SOH\ast N_{cycles}$$

(40)

where $SOH$ refers to the state of the health of the storage, and $N_{cycles}$ refers to maximum number of cycle up to which the storage will last.

Therefore, the availability cost the battery energy storage system can be calculated by the following equation:

$$c_{bess}={\displaystyle \frac{LCCOES}{E_{out}}}$$

(41)

The cost of charging and discharging is determined by the availability cost ($c_{bess}$) and the efficiency $({\eta }_{bess}$) of the battery, as follows:

$$c_{chr}=c_{bess}{\eta }_{bess}$$

(42)

$$c_{dich}={\displaystyle \frac{1}{{\eta }_{bess}}}{c}_{bess}$$

(43)

Table 2. Parameters for Li-ion batteries.

Parameters	Value
$i$ (%)	7
$y$ (years)	20
$c_s$ (AUD/kWh)	1272
${DOD}_{max}$ (%)	90
$c_{pc}$ (AUD/kW)	740
$c_{pd}$ (AUD/kW)	128
$c_{fixed}$ (AUD/kW)	11
$V_{om}$	0.03
$B_{elec}$ (AUD/kWh)	0.16
$d$ (day)	365
$h_d$ (hour)	2
$V_{elec}$	0.03
$SOH$ (%)	80
$N_{cycles}$ (cycles)	6000

provides the cost and technical parameters to calculate the availability cost, which can be found from [39] for Li-ion batteries.

4. Simulation Results

To verify the efficiency of the proposed solution methodology for optimal scheduling of battery energy storage system in radial distribution systems, an Australian distribution network is employed. The electrical configuration of this system is presented in Figure 2. The system has a 11 kV voltage at the substation bus and 415 V at the distribution side. The capacity of the transformer is 200 kVA. To connect the buses, a moon conductor, which is of seven 4.75 mm AAC strands, has been used. A total of 55 houses are connected to the system, and PV is also connected to the buses where load is connected. Each PV system is equipped with 6 kWp. The peak load is 155 kW. The battery capacity is 200 kWh. The minimum and maximum state of charge is limited to 0.2 to 1. The time resolution of the load and the PV power is 1 h.

To construct the solar radiation profile by using Equation (1) to Equation (11), the parameter values used are $r_o$ = 0.95; $r_1$ = 0.99; $r_k$ = 1.02. The solar radiation profile is shown in Figure 3.

A detailed Australian residential load demand has been constructed for 2024 and onwards [40]. The load profile specifically tailored for 2024 has been employed in the study, serving as the basis for simulations. Figure 4 shows the load profile for 24 h period.

The cost of charging and discharging can be calculated from Equations (42) and (43), which are $c_{chr}$ = 0.415 AUD/kWh and $c_{dich}=$ 0.046 AUD/kWh.

Table 3. Time-of-use tariff structure for Australian states.

Queensland (QLD)			South Australia (SA)
Time-of-Use (TOU)	Designation	Energy prices (AUD/kWh)	Time-of-Use (TOU)	Designation	Energy prices (AUD/kWh)
19:00–9:00	Shoulder	0.3012	21:00–10:00	Shoulder	0.3107
9:00–16:00	Off-peak	0.2756	10:00–16:00	Off-peak	0.2952
16:00–19:00	On-peak	0.4365	16:00–21:00	On-peak	0.5159
Victoria (VIC)			Western Australia (WA)
Time-of-Use (TOU)	Designation	Energy prices (AUD/kWh)	Time-of-Use (TOU)	Designation	Energy prices (AUD/kWh)
21:00–10:00	Shoulder	0.2500	21:00–07:00	Off-peak	0.0841
10:00–15:00	Off-peak	0.1837	07:00–15:00	On-peak	0.5153
15:00–21:00	On-peak	0.3032	15:00–21:00	Shoulder	0.2311
Tasmania (TAS)			Australian Capital Territory (ACT)
Time-of-Use (TOU)	Designation	Energy prices (AUD/kWh)	Time-of-Use (TOU)	Designation	Energy prices (AUD/kWh)
21:00–07:00	Off-peak	0.1669	22:00–07:00	Off-peak	0.2017
07:00–10:00	On-peak	0.3584	07:00–09:00	On-peak	0.3789
10:00–16:00	Off-peak	0.1669	09:00–17:00	Shoulder	0.2761
16:00–21:00	On-peak	0.3584	17:00–20:00	On-peak	0.3789
			20:00–22:00	Shoulder	0.2761
New South Wales (NSW)			Norther Territory (NT)
Time-of-Use (TOU)	Designation	Energy prices (AUD/kWh)	Time-of-Use (TOU)	Designation	Energy prices (AUD/kWh)
21:00–15:00	Off-peak	0.2315	18:00–06:00	Off-peak	0.2627
15:00–21:00	On-peak	0.5562	06:00–18:00	On-peak	0.3445

presents the energy-pricing structure for different states of Australia broken down by three time-of-use (TOU) periods, such as on-peak, off-peak, and shoulder. The on-peak period refers to the times when the tariff rate is expensive due to the high electricity demand. Off-peak times represent the periods of the low electricity demand, which results in lower tariff rates. Shoulder time periods fall between on-peak and off-peak and the pricing is higher than off-peak and lower than on-peak. A flat tariff refers to a pricing structure where the rate charged for a service or commodity is fixed and does not vary based on consumption or usage. This means the customer pays the same amount regardless of how much of the service or product they use. A flat tariff rate of 0.1837 AUD/kWh has been used in this study.

Figure 5 illustrates the daily power balance and energy storage dynamics for various Australian states, showcasing the relationship between PV generation, energy storage charging and discharging, and grid interaction. All states import power from the grid during late night and early morning. Energy storage becomes charged during late night or early morning due to the off-peak tariff in WA, TAS, NT, NSW, and ACT. Again, battery becomes discharged before the peak PV period for these states. This pre-emptive discharging ensures that energy storage systems are not competing with PV energy production for grid support. Instead, they help stabilize the grid during the morning peak demand and recharge during the day when solar generation is available. For states like QLD, VIC, and SA, the optimization does not find it cost-effective to charge the battery during early hours as the tariff system is shoulder for these states. The energy prices during this period are not as low as off-peak hours, making it more expensive to charge the batteries at this time. For all the states, the battery starts to charge at 10 a.m. and reaches the peak storage level during the PV hours. Again, the battery becomes discharged during high load demand to support the grid. In a flat tariff system, there is no incentive to charge the battery during specific times of the day based on cost. Therefore, the charging of energy storage typically occurs during the solar generation period. During times of high demand, when grid usage is at its highest, energy stored in the batteries is released to support the grid. This helps reduce strain on the grid by providing additional supply from the battery system, which is especially important during times of high energy consumption. Discharging the battery during these periods allows the system to avoid importing power from the grid, which is more expensive.

Table 4. Energy matrices for Australian states.

Parameter	ACT	NSW	NT	SA	TAS	VIC	WA	QLD	Flat Tariff
Daily imported energy (kWh)	1134	1219	1219	1112	1168	1146	1253	1146	1146
Daily exported energy (kWh)	585	678	675	571	618	605	709	7605	602
Utility bill (AUD)	599	747	659	785	571	550	611	668	396
Optimized bill (AUD)	144	257	126	280	175	202	134	200	114
Savings (AUD)	455	490	533	505	396	348	477	468	282

presents a comparison of various states and a flat tariff system based on different energy-related metrics, including daily imported energy, daily exported energy, utility bill, and optimized bill. The utility bill varies across different states ranging from 550 AUD in VIC to 785 AUD in SA due to the grid prices and tariff structure. The utility bill shows significant reductions across all states after applying optimization. Using the current tariff structure, ACT can save up to 455 AUD, NSW 490 AUD, NT 533 AUD, SA 505 AUD, TAS 396 AUD, VIC 348 AUD, WA 477 AUD, QLD 468 AUD, and 282 AUD using the flat tariff. This suggests that, for consumers who can optimize their energy use and manage it according to the peak and off-peak rates, the current tariff structure might lead to higher savings. However, for those who prefer simplicity or have a more consistent usage pattern, the flat tariff might be an easier but slightly less cost-saving option.

From the above discussion, it is evident that the battery is charged during solar generation hours and discharged during peak demand hours. Consequently, the voltage profiles during these periods are influenced by the charging and discharging dynamics of the battery. As an illustrative case, the Queensland voltage profile for all buses at 11:00, 12:00, 13:00, 20:00, and 21:00 h is shown in Figure 6. Before optimization, the voltages at buses 6, 7, and 8 at 11:00 a.m. are 1.0542 p.u., 1.0512 p.u., and 1.0526 p.u., respectively. After the optimization shown in Figure 7, utilizing the battery, the voltage profile improves significantly by 0.728% for these buses. Similarly, during the peak demand period at 8:00 p.m., the voltages at these buses are 0.9468 p.u., 0.9496 p.u., and 0.9484 p.u., respectively. As the battery discharges during this time, the voltages at these buses rise to 0.9569 p.u., 0.9596 p.u., and 0.9585 p.u., respectively, demonstrating a notable improvement in voltage stability during high-demand periods.

5. Conclusions

This study proposes a robust and innovative approach to optimizing energy management systems by employing a mixed-integer linear programming (MILP) framework. The primary objectives are to minimize utility bills and maintain grid stability, while addressing key economic and technical challenges posed by renewable energy integration. Real-time weather data are extracted directly from a reliable weather-forecasting website, using web-scraping techniques. This eliminates the need for manual data collection or dependency on outdated static datasets. Automating this process bridges the gap between PV generation forecasting and real-world weather conditions, ensuring the use of up-to-date and accurate weather data. The clear sky model is applied to convert the extracted weather data (e.g., solar irradiance) into a solar generation profile, accounting for geographical and meteorological variations. This enhances the reliability of PV generation estimates, which are critical for day-ahead energy management and optimization. This study evaluates both time-of-use (TOU) and flat tariff systems across eight Australian states, representing real-world economic and regulatory conditions. TOU tariffs divide electricity prices into peak, off-peak, and shoulder periods, encouraging load and storage optimization to reduce costs during high-price periods. Flat tariffs offer a consistent rate, enabling straightforward cost analysis but limiting load-shifting opportunities. The economic analysis demonstrates how different tariff structures influence the optimal scheduling of battery energy storage systems (BESS). BESS operations, such as charging during low-cost periods and discharging during high-cost periods, are tailored to maximize savings under the given tariff system. Voltage constraints are embedded in the MILP model to ensure that the system operates within permissible voltage limits, avoiding issues like voltage drops or over-voltages that could destabilize the grid. High penetration of renewable energy, especially distributed PV systems, can cause voltage fluctuations. The proposed framework addresses these challenges by maintaining a stable voltage profile across the distribution network. By combining economic and technical objectives, the model provides a holistic solution for energy management, making it directly applicable to real-world systems. The proposed framework equips utility operators with actionable insights to optimize energy storage systems, improve grid performance, and adapt to dynamic market conditions.