Enhanced Microgrid Control through Genetic Predictive Control: Integrating Genetic Algorithms with Model Predictive Control for Improved Non-Linearity and Non-Convexity Handling

Abstract

Microgrid (MG) control is crucial for efficient, reliable, and sustainable energy management in distributed energy systems. Genetic Algorithm-based energy management systems (GA-EMS) can optimally control MGs by solving complex, non-linear, and non-convex problems but may struggle with real-time application due to their computational demands. Model Predictive Control (MPC)-based EMS, which predicts future behaviour to ensure optimal performance, usually depends on linear models. This paper introduces a novel Genetic Predictive Control (GPC) method that combines a GA and MPC to enhance resource allocation, balance multiple objectives, and adapt dynamically to changing conditions. Integrating GAs with MPC improves the handling of non-linearities and non-convexity, resulting in more accurate and effective control. Comparative analysis reveals that GPC significantly reduces excess power production, improves resource allocation, and balances cost, emissions, and power efficiency. For example, in the Mutation–Random Selection scenario, GPC reduced excess power to 76.0 W compared to 87.0 W with GA; in the Crossover-Elitism scenario, GPC achieved a lower daily cost of USD 113.94 versus the GA’s USD 127.80 and reduced carbon emissions to 52.83 kg CO2e compared to the GA’s 69.71 kg CO2e. While MPC optimises a weighted sum of objectives, setting appropriate weights can be difficult and may lead to non-convex problems. GAs offer multi-objective optimisation, providing Pareto-optimal solutions. GPC maintains optimal performance by forecasting future load demands and adjusting control actions dynamically. Although GPC can sometimes result in higher costs, such as USD 113.94 compared to USD 131.90 in the Crossover–Random Selection scenario, it achieves a better balance among various metrics, proving cost-effective in the long term. By reducing excess power and emissions, GPC promotes economic savings and sustainability. These findings highlight GPC’s potential as a versatile, efficient, and environmentally beneficial tool for power generation systems.

1. Introduction

Microgrids (MGs) play a crucial role in contemporary energy systems as they consist of distributed energy resources and loads, guaranteeing the efficient, dependable, and sustainable distribution of energy [1,2]. This is particularly significant in situations where traditional power systems encounter obstacles, such as in remote or isolated regions [3]. Efficient control and energy management in MGs is essential for effectively distributing resources between supply and demand, resulting in reduced operational expenses and environmental effects [4,5]. Conventional control approaches encounter challenges because of the intrinsic nonlinearity and nonconvexity of MG systems [6,7].

Ref. [8] investigates a power control technique for inverter-based distributed generation (DG) in an islanded MG to improve voltage, frequency control, and dynamic response. The method employs an internal current control loop and an external power control loop using a synchronous reference frame with a PI controller based on voltage-frequency (VF) control. To optimise controller parameters, an intelligent search technique combining Particle Swarm Optimisation (PSO), and a Genetic Algorithm (GA) is used. Simulation results demonstrate improved power quality under varying load and islanded conditions. Ref. [9] proposes a strategy using an automatic commutation switch (ACS) and a GA to balance unbalanced DC loads in a bipolar DC distribution network. The method adjusts power supply polarity to reduce power losses and improve power quality. Simulations in MATLAB/Simulink confirm the approach’s effectiveness. GAs have been utilised to swiftly discover optimal solutions and tackle very intricate issues that may be unsolvable using conventional mathematical approaches. GAs excel at finding global optimal solutions and are capable of handling complex, non-linear, and non-convex problems [10,11]. Although conventional genetic algorithms have strong optimisation capabilities, they can be computationally demanding and may encounter difficulties in real-time control applications [12]. On the other hand, Model Predictive Control (MPC) is often favoured due to its inherent capacity to forecast future actions within a specified timeframe, while simultaneously ensuring optimal performance and compliance with restrictions [13]. However, MPC typically operates with linear models, which limits its effectiveness in controlling the nonlinear dynamics of MGs [14].

This work introduces a novel approach to controlling MGs, known as genetic predictive control (GPC). The integration of a GA with MPC improves the allocation of resources, the balance of numerous objectives, and the responsiveness to dynamic changes. By combining the strong optimisation capabilities of GAs with the predictive capability of MPC, it becomes possible to develop a control approach that is significantly more precise and effective. The suggested use of GAs in MPC enhances the ability to manage nonlinearities and non-convexities in the system being analysed.

The subsequent sections of the paper are structured in the following manner. The paper begins with a comprehensive examination of the literature to identify the most efficient GAs and the challenges associated with MPC. This is followed by a detailed explanation of the methodology used to implement GPC. The subsequent section presents the results of a comparative analysis, demonstrating the enhanced performance achieved using GPC-based energy management systems (EMSs). Finally, the paper concludes with a discussion of the findings and potential avenues for future research.

1.1. Literature Review

The energy management of MGs has been a highly researched field, with a growing focus on developing an EMS that is efficient, independent, and environmentally friendly. Several EMSs have been suggested and put into practice, each having its advantages and disadvantages in terms of their performance and the problem formulation.

GAs are a type of evolutionary algorithm that imitates the process of natural selection. Due to the implementation of natural selection in GAs, it is anticipated that they possess the potential to effectively solve highly nonlinear, nonconvex, and difficult optimisation problems [15,16]. GAs have been effectively utilised for optimising MG control in several settings, such as:

1. Energy Dispatch Optimisation: A GA is utilised to optimise the distribution of energy for DERs to decrease operational costs and carbon emission, while ensuring the satisfaction of energy demand [17,18]. This research has demonstrated the efficacy of GAs in achieving or approaching near-global optimal solutions in cases where traditional methods are unable to owing to the complexity of the problem space, as observed in multiple investigations [19,20].

2. Multi-Objective Optimisation: Multi-objective optimisation is a problem-solving technique that uses GAs to find solutions that are Pareto-optimal. These solutions can balance numerous conflicting objectives. This capability is significant for managing MGs; cost, carbon emission, and power efficiency trade-offs must be addressed [21,22]. The existing literature has confirmed that achieving a balance among these three objectives leads to effective solutions and a wide range of alternatives that are valuable for decision makers [23,24].

Nevertheless, the computational demand of GAs can be quite high, particularly in real-time applications [25]. The iterative nature of GAs, which includes selection, crossover, and mutation processes, can be computationally demanding and significantly restricts its use in settings requiring fast responses [26].

Recent research has been focused on finding ways to improve the effectiveness of GAs in MG applications. For example, advanced algorithms like the NSGA-II have been used to optimise EMS for an MG by considering many objectives related to cost and emission functions [27,28,29]. Another development involves the integration of GAs with other optimisation approaches to achieve enhanced performance [30].

MPC is a control approach that uses a system model to forecast future states and aims to find the best control actions within a defined prediction horizon. MPC is widely applied in MG control, as it not only incorporates standard prediction and optimisation capabilities, but also effectively addresses constraints and continuously optimises performance [31]. Ref. [32] presents an intraday robust energy management framework for multi-microgrids (MMGs) using distributed dynamic tube model predictive control (DD-TMPC) to handle uncertainties in renewable energy and load demand. The approach reduces conservativeness and enhances economic efficiency by decomposing the uncertainty problem into two deterministic, lower-complexity scheduling problems. A game theory-based energy trading mechanism is also introduced to balance robustness and economy. Simulations confirm the effectiveness of the proposed strategy. This study in [33] addresses the stable operation of integrated energy systems (IES) amid fluctuations in heat or electricity loads, energy or communication issues, and interactions between different energy networks. A distributed model predictive control (DMPC) consensus algorithm, based on second-order cone programming (SOCP), is proposed to ensure stability by considering dynamic behaviours, energy coupling, and multi-time scale constraints. The approach achieves optimal regulation of frequency, voltage, temperature, pressure, and flow, and is validated through various simulation scenarios, including load and energy flow variations, node removals, and topology changes. Recent advancements in MG optimisation include the development of hybrid intelligent control systems that integrate rule-based control and deep learning techniques. For instance, a hybrid intelligent control system for adaptive MG optimisation that integrates rule-based control and deep learning techniques has been proposed, demonstrating significant improvements in the adaptive control of MGs [34]. Another approach involves a hybrid method based on a logic predictive controller for flexible hybrid MGs with plug-and-play capabilities, enhancing the adaptability and efficiency of MG management [35]. Furthermore, the Switched Auto-Regressive Neural Control (S-ANC) method has been developed for the energy management of hybrid MGs, showcasing effective energy optimisation and management [36]. In addition, a comprehensive review of MG energy management strategies from the energy trilemma perspective highlights the importance of balancing energy security, environmental sustainability, and economic viability in MG operations [37]. These strategies emphasise the need for flexible and adaptive control methods to meet the dynamic demands of MG systems [38].

The primary uses of MPC in MG control are as follows:

1. Predictive Energy Management: MPC anticipates future energy demands and adjusts control actions to minimise operational costs, reduce emissions, and ensure dependable operation [39]. Several researchers have conducted studies and provided evidence of the efficacy of MPC in preserving optimal performance in dynamic contexts [40].

2. Managing Uncertainties: Incorporating forecasting and constraints into the optimisation process allows MPC to effectively handle uncertain fluctuations in renewable generation and demand load changes [41]. This capability is particularly crucial in MGs with high penetration levels of renewable generation [42].

Nevertheless, a significant limitation of MPC is its reliance on linear models. The majority of MG systems demonstrate nonlinear behaviour, which linear models are unable to accurately define [43]. This scenario can result in a decline in performance and a decrease in the accuracy of control activities [29]. Many advances have been implemented to address this disadvantage and advanced optimisation techniques incorporated into the MPC framework. For instance, several adaptive MPC strategies have been developed to modify the control model using real-time data to enhance the accuracy and efficacy of MG control [44]. However, this new variation of MPC does not address the issue of dealing with non-linear models. For these reasons, the objective of this work is to integrate GAs with MPC to leverage the advantages of both approaches while mitigating their limitations. The GPC approach enhances the control of MG by combining the robust optimisation characteristics of GAs with the predictive capabilities of MPC.

1.2. Contributions of This Paper

• Introduction of a novel GPC method: This paper introduces a new control method, GPC, which integrates GAs with MPC to enhance the management of MGs. The integration leverages the global optimisation capabilities of GAs with the predictive power of MPC, addressing the limitations of each individual method.
• Enhanced handling of non-linear and non-convex problems: The GPC method is specifically designed to handle the complex, non-linear, and non-convex characteristics of MG systems more effectively than conventional approaches. By incorporating GAs within the MPC framework, the proposed method better manages the inherent complexities and dynamic behaviours of MGs.
• Dynamic adaptability to changing conditions: GPC improves the adaptability of MG control by dynamically adjusting control actions based on forecasted future load demands and system conditions. This ability allows the MG to maintain optimal performance despite fluctuating supply and demand, which is essential for reliable and efficient operation.
• Contribution to hybrid control strategies: The integration of GAs and MPC in the proposed GPC method contributes to the development of hybrid control strategies, combining different optimisation techniques to achieve better performance. This approach could be extended to other domains beyond MGs, offering a framework for future research in advanced control methods.

Unlike purely GA-based methods, our GPC approach leverages the predictive capabilities of MPC to anticipate future system states and adjust control actions accordingly. This integration enhances the robustness and efficiency of MG operations by dynamically balancing multiple objectives, including cost, emissions, and power generation alignment. The following sections provide a detailed description of the GPC implementation process and evaluate its performance across various scenarios, highlighting the significant improvements over traditional GA and MPC methods.

2. Methodology of Building and Implementing GPC

This study proposes a solution utilising GPC, which integrates the advantageous characteristics of GAs and MPC through the implementation of clearly defined methodologies for MG control and optimisation. The technique will consist of establishing the control objective, constructing a model of the MG system, developing the GA and MPC subsystems, integrating the GA with MPC, and simulating the GPC implementation. The next subsections provide a comprehensive discussion of each phase and the corresponding equations that need to be implemented inside the GPC framework.

The primary control objectives of GPC for the MG are:

• Minimising excess power production: Optimise power generation to closely match load demand, with the aim of minimising overgeneration and consequently minimising the power injection in the upstream network.
• Balancing cost and emissions: Achieve a balance between cost and carbon emissions to maximise the economic and environmental viability of the MG.

2.1. MG System Model

Figure 1 shows a schematic diagram of the overall power balancing in an MG system and describes the main power flows between major components. The total generated power is composed of multiple DERs, renewable generators like photovoltaic (PV) panels and wind turbines (WTs) and conventional ones. It is shown that the power generated by DERs is fed into the system to supply the load demand, representing the power needed by consumers inside the MG. Utility grid power is the general surplus power produced more than load demand and can, therefore, be stored in energy storage systems (ESSs) or curtailed. Additionally, it considers the loss of power through transmission, distribution, and inefficiencies in conversion within a MG. The need for effective EMS and advanced control systems is highlighted by this comprehensive depiction to ensure the reliable and sustainable operation of MGs.

This diagram illustrates in Figure 1 the role that the EMS will play in coordinating and optimising the multiple operations occurring within this MG. The EMS integrates all sources of information to manage power flows and assure efficient distribution of energy. This complete diagram shows the need for an effective EMS and sophisticated control systems for the reliable and sustainable operation of MGs.

The power balancing equation for the MG is expressed as follows:

$$P_{gen}\left(k\right)=P_{PV}\left(k\right)+P_{WT}\left(k\right)$$

(1)

$$P_{gen}\left(k\right) + P_{dis}\left(k\right) = P_{load}\left(k\right) + P_{char}\left(k\right) + P_{loss}\left(k\right) + P_{grid}\left(k\right)$$

(2)

where $P_{PV}\left(k\right)$ and $P_{WT}\left(k\right)$ are the power outputs from PV, and WT, respectively. $P_{char}$ and $P_{dis}$ are the charging and discharging power of ESS, respectively. $P_{gen}\left(k\right)$ represents the total power generated by DERs, $P_{load}\left(k\right)$ is the load demand, and $P_{grid}\left(k\right)$ is the power from/to the utility grid. The inclusion of ESS impacts the equation by allowing excess power to be stored when available and discharged to meet load demand or balance supply during periods of low generation. $P_{loss}\left(k\right)$ represents the power losses in the system at the instant time $k.$

2.2. The Implementation of MPC

To design the MPC system for the MG described, we need to define the control vector, state vector, and output vector. These vectors are crucial for formulating the MPC problem, which optimises the MG operation over a prediction horizon.

• The control vector $u\left(k\right)$ consists of the variables that the MPC can manipulate to achieve the desired performance.

  $$u\left(k\right)=[P_{gen}\left(k\right);P_{char}\left(k\right);P_{dis}\left(k\right)]$$

  (3)
• The state vector $x\left(k\right)$ includes variables that represent the current status of the system.

  $$x\left(k\right)=\left[SOC\left(k\right)\right]$$

  (4)
• The output vector $y\left(k\right)$ consists of the variables that the MPC aims to regulate or track.

$$y\left(k\right)=[P_{loss}\left(k\right);P_{grid}\left(k\right)]$$

(5)

To define the state-space representation of the system for MPC, we need to establish the matrices $A$, $B$, $C,$ and $D$ describe the dynamics and relationships within the MG. These matrices relate the state vector, control vector, and output vector.

$$\begin{gathered} x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right) \\ y\left(k\right)=Cx\left(k\right)+Du\left(k\right) \end{gathered}$$

(6)

The $A$ matrix represents the relationship between the current state and the next state. For the MG system, it includes how the state of charge (SOC) of the ESS evolves over time.

$$A=1$$

(7)

The $B$ matrix represents the influence of the control inputs on the state vector. For our MG, the control inputs are the charging and discharging powers of the ESS, and the power exchange with the utility grid.

$$B=\left[\begin{array}{ccc}{\eta }_{char}& {-\eta }_{dis}& 0\end{array}\right]$$

(8)

The $C$ matrix maps the state vector to the output vector. It includes how the states contribute to the outputs, which, in our case, are the power losses, and the power exchanged with the grid.

$$C=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$$

(9)

The $D$ matrix directly relates the control inputs to the outputs. Given the definitions of the output vector components, this matrix includes the direct contributions of the control inputs:

$$D=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 1& 0& 1\end{array}\right]$$

(10)

The main task of MPC is to cope with the non-linearity in this paper. It can be clearly explained as follows:

• For the PV panel and WT, the power output $P_{PV}$ and $P_{WT}$ can be modelled using the following equation [45]:

  $$\begin{gathered} P_{PV}\left(k\right)=f\left(\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e},\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\right)={\eta }_{PV}.A_{PV}.irradiance.(1-{\beta }_{PV.}temperature-T_{ref}) \\ f\left(x\right)=\left\{\begin{array}{cc}\hfill 0,& \mathrm{if} wind speed\le v_{cu{t}_{in}}orwind speed>v_{cut\_out}\hfill \\ \hfill {\displaystyle \frac{1}{2}}\rho A_{WT}{C}_P{(wind speed)}^3,& \mathrm{if} v_{cu{t}_{in}}\le wind speed\le v_{rated}\hfill \\ \hfill P_{rated},& \mathrm{if} v_{cu{t}_{rated}}\le wind speed\le v_{cut\_out}\hfill \end{array}\right. \end{gathered}$$

  (11)

  where $g\left(\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\right)$ is a non-linear function. ${\eta }_{PV}$ is the efficiency of the PV system. $A_{PV}$ is the area of the PV panels. ${\beta }_{PV}$ is the temperature coefficient of power (°C⁻¹). $T_{ref}$ is the reference temperature, typically 25 °C. $\rho$ is the air density (1.225 kg/m³). $A_{WT}$ is the swept area of the WT blades (in m²). $C_P$ is the power coefficient of the WT (a dimensionless value, typically around 0.4 to 0.5). $v_{cut\_in}$ is the cut-in wind speed (the minimum wind speed at which the turbine generates power). $v_{cut\_out}$ is the cut-out wind speed (the wind speed at which the turbine stops to prevent damage). $v_{rated}$ is the rated wind speed (the wind speed at which the turbine generates its rated power). $P_{rated}$ is the rated power output of the WT.
• Power losses in transmission and distribution lines can be expressed as [46]

  $$P_{loss}\left(k\right)=I^{2}R$$

  (12)

  where $I$ is the current and $R$ is the resistance of the line. $I^2$ is the non-linear value in Equation (12).
• The $SOC$ of the ESS and its charging/discharging efficiency can be modelled as [35]

  $$SOC\left(k+1\right)=SOC\left(k\right)+{\eta }_{char}{P}_{char}-{\eta }_{dis}{P}_{dis}$$

  (13)

  where ${\eta }_{char}$ and ${\eta }_{dis}$ are non-linear functions of the charging and discharging power.

Charging and discharging for the ESS cannot happen simultaneously, as is implied by the following formula:

$$P_{char}\left(k\right)P_{dis}\left(k\right)\le 0$$

(14)

For the charging, the constraint can be written as

$$P_{char}\left(k\right)\le 0\hspace{1em}\hspace{1em}\hspace{1em}{P}_{char}\left(k\right)\ge 0$$

(15)

For the discharging, the constraint can be written as

$$P_{dis}\left(k\right)\le 0\hspace{1em}\hspace{1em}\hspace{1em}{P}_{dis}\left(k\right)\ge 0$$

(16)

Equations (15) and (16) provide constraints on the charging and discharging power of the ESS to ensure that charging and discharging do not occur simultaneously.

• $P_{char}\left(k\right)\le 0$: This inequality requires that the charging power be zero or negative. Physically, this means that when the ESS is charging, it absorbs energy from the grid or other sources. A value of $P_{char}\left(k\right)=0$ indicates no charging, while $P_{char}\left(k\right)<0$ means the ESS is actively being charged.
• $P_{char}\left(k\right)\ge 0$: This constraint implies that charging power should be zero or positive, stating that charging power cannot be negative. When both conditions are applied, they confirm that during charging, the power into the system cannot exceed the maximum charge rate.
• $P_{dis}\left(k\right)\le 0$: This inequality states that the discharging power should be zero or negative. Physically, this means that when the ESS is discharging, it releases energy (power) back into the grid or the connected load. If $P_{dis}\left(k\right)=0$, no discharging occurs; if $P_{dis}\left(k\right)<0$, discharging is happening, and the ESS is supplying power.
• $P_{dis}\left(k\right)\ge 0$: This constraint indicates that discharging power should be zero or positive, effectively stating that discharging power cannot be negative. Together with the previous condition, these two inequalities imply that during discharging, the power drawn by the system cannot exceed the maximum discharge limit (which would be indicated as zero or more).

MPC optimises control inputs by solving a constrained optimisation problem over a finite prediction horizon, $N_P$. The general form of the MPC optimisation problem can be expressed as

$$\underset{u\left(k\right)}{\min }\sum _{m=0}^{N_P}J\left(x\left(k+m\right),u\left(k+m\right)\right)$$

(17)

$$\begin{gathered} subject to: \\ x\left(k+m+1\right)=f\left(x\left(k+m\right),u\left(k+m\right)\right), \end{gathered}$$

(18)

$$x\left(k+m\right)\in \mathit{X}$$

(19)

$$u\left(k+m\right)\in \mathit{U},$$

(20)

where $J$ is a given cost function, $N_P$ is a given number of prediction steps, $f$ is the system dynamics, and $\mathit{X}$ and $\mathit{U}$ are the sets of admissible states and control inputs, respectively.

The cost function $J$ typically has terms that trade off against tracking performance, control effort, and constraint violations. For the specific MG control problem, the cost function $J$ can be defined as

$$J=\sum _{m=0}^{N_P}\left(w_1{\left(P_{gen}\left(k+m\right)-P_{load}\left(k+m\right)\right)}^2+w_{2}C\left(k+m\right)+w_{3}E\left(k+m\right)\right)$$

(21)

where $w_1,$ $w_2$, and $w_3$ are weighting factors, $C\left(k\right)$ represents the cost of power generation, and $E\left(k\right)$ represents the emissions. From Equation (2), $P_{grid}$ is minus $P_{gen}$ and ($P_{load}+P_{loss}+P_{ESS})$. This can be negative or positive. Equation (21) is likely derived from the MPC, a control strategy commonly used in power systems for optimal energy management. In MPC, the objective function $J$ represents a cumulative cost over a future prediction horizon $N_P$. The MPC algorithm optimises control actions by predicting future states and minimising the cost function subject to system constraints.

• The term ${{(P}_{gen}-P_{load})}^2$ aims to minimise the power imbalance between generation and load, ensuring stable and reliable operation.
• The term $C(k+m)$ could represent the financial costs associated with energy generation, procurement, or operation.
• The term $E(k+m)$ likely reflects environmental costs, such as emissions, which are becoming increasingly important in sustainable energy management.

The constraints for the MPC problem include the following:

• Power Balance Constraint:

  $$P_{balance}\left(k\right)={\sum }_{i=1}^{N_{DER}}{P}_i\left(k\right)=P_{load}(k) + P_{char}\left(k\right) + P_{loss}\left(k\right) + P_{grid}\left(k\right)-P_{dis}\left(k\right)$$

  (22)

  where $P_i\left(k\right)$ is the power generated by the $i$-th DER and $N_{DER}$ is the number of DERs.
• Generation Capacity Constraints:

  $$P_{i,min}\le P_i\left(k\right)\le P_{i,max}$$

  (23)

  where $P_{i,min}$ and $P_{i,max}$ are the minimum and maximum power generation limits of the $i$-th DER.
• Ramp Rate Constraints:

  $${-R}_i\le P_i\left(k\right)-P_i\left(k-1\right)\le R_i$$

  (24)

  where $R_i$ is the ramp rate limit of the $i$-th DER.
• Emissions Constraint:

  $$E\left(k\right)=\sum _{i=1}^{N_{DER}}{E}_i\left(k\right)\le E_{max}$$

  (25)

  where $E_i\left(k\right)$ is the emission produced by the $i$-th DER and $E_{max}$ is the maximum allowable emission.

2.3. The Implementation of the GA Algorithm

GAs finds a optimal solution $u\left(k\right)$ to control inputs in MG by evolving a population of candidate solutions. Selection, crossover, and mutation are automated optimisation processes in GAs for MG control. The fitness function of GAs in MG control is expressed as the inverse of the cost function $J$:

$$Fitness\left(u\right)={\displaystyle \frac{1}{J\left(x,u\right)}}$$

(26)

The GA process is summarised by the following steps:

• Initialisation: An initial population of candidate solutions, chromosomes in the form of control inputs, is created.
• Selection: Parent chromosomes are selected and survive according to computed fitness values for each chromosome.
• Crossover: Parents chromosomes are combined to produce the offspring.
• Mutation: Small random changes in chromosomes of offspring are introduced to provide an element of randomness and to retain diversity.
• Evaluation: The fitness of offspring chromosomes is computed.
• Replacement: A new population is created considering the best chromosomes of the current population and offsprings.

The key operations in GA are defined as following functions:

• Crossover: A single-point crossover operation can be defined as

  $$\begin{gathered} {Offspring}_1=\left({Parent}_1\left[: m\right], {Parent}_2\left[m :\right]\right) \\ {Offspring}_2=\left({Parent}_2\left[: m\right], {Parent}_1\left[m :\right]\right) \end{gathered}$$

  (27)

  where $m$ is the crossover point.
• Mutation: A mutation operation can be defined as

  $$Offspring\left[i\right]=Offspring\left[i\right]+∆$$

  (28)

  where $∆$ is a random perturbation.

2.4. The Implementation of GPC

The GPC method integrates GAs with MPC by using a GA to optimise the control inputs over the MPC prediction horizon. The integrated GPC formulation is developed as follows:

• Prediction: Use system model to predict future states over the prediction horizon.
• Optimisation: Apply GA to optimise the control inputs $u\left(k\right)$ over the prediction horizon.
• Implementation: Implement the optimised control inputs in the MG system.
• Repetition: Execute repetition of the process at each of the control steps to adapt to the changing conditions.

The GA solves the MPC problem formulated by Equations (10)–(14).

The GPC method was developed using Python. Basically, the simulation model is with the DER model at a higher level of details along with the models of the load profiles and the distribution power components. Its actual working has been compared with the conventional methods of MPC and GAs and further assessed under several scenarios to evaluate its real-time efficacy to achieve the set control objectives.

The key metrics used for evaluation include three factors:

• Excess Power Production: Measured as the total surplus power generated beyond the load demand, and the power to be stored in the ESS. This power will be injected into upstream grid.
• Power Generation Costs: Calculated as an estimate of the operational costs for DERs.
• Emissions: Quantified in terms of the total emissions produced by the MG.

The simulation results showed that the GPC-based method considerably decreases excess power production, increasing the efficiency of resource allocation and balancing multiple objectives much better than the conventional methods based on MPC and GAs.

2.5. Practical Implementation Steps of GPC

The implementation of GPC in a MG follows a well-defined process so that the developed control presents efficiency and accuracy. As shown in Figure 2, the process begins by defining all system components, including DERs, loads, and storage systems. Control objectives are then outlined-some of the common objectives are minimisation of excess power production, cost versus emissions trade-off, and maximization of overall power efficiency.

In this, the next step is the initialization of parameters for both the GA and MPC, including the population size, crossover rate, and mutation rate; in addition to the number of generations for the GA; and the prediction horizon, control horizon, and weighting factors for the cost function for the MPC.

As soon as these parameters are set, an initial work in the GA begins, where it first reads the current state of the MG to get its operational status and power levels. An MPC predicts future states over the prediction horizon, followed by the initialization of the population of chromosomes, each representing potential control actions in Figure 2.

The core of the GPC process lies in the iterative loop of the GA. For a specified number of generations, parent chromosomes are selected based on their fitness values. The fitness of each chromosome is evaluated through the calculation of the cost function, and fitness is inversely related to this cost. During crossover, pairs of parent chromosomes exchange genetic information at a specified crossover point, producing offspring. Mutation makes some genes change randomly so genetic diversity can be retained. Then the offspring replace part of the adult population with new generation.

The control actions are considered for only implementing the chromosome that contains the best value, in other words, the one with the highest fitness. Its control inputs are then applied to the MG. The system performance is tracked to determine whether that new control action has provided improvement. If not, the system model is updated based on feedback, and the process is repeated. If performance has improved, the process concludes. This structured design of GPC is highly effective in managing the non-linearity and non-convexity of the MG system by leveraging the optimization capabilities of GAs within the MPC framework. It ensures efficient, accurate, and robust control.

In a scenario where both wind power and PV generation are zero, the performance of the proposed GPC method for managing the MG will depend on the capacity and SOC of the ESS.

Performance of the Proposed Controller:

1.

ESS Support for Load Demand:

○

When there is no power generation from wind or PV sources, the ESS becomes the primary source of electricity for meeting the load demand. The GPC method is designed to optimise the use of the ESS by dynamically adjusting its charging and discharging cycles based on forecasted future load demands and the current SOC of the ESS.

○

If the ESS has a sufficient SOC (meaning it has enough stored energy), the GPC will discharge power from the ESS to support the load demand. The control strategy ensures that the ESS discharges at an optimal rate to meet the load requirements while also minimising power losses and maintaining efficient operation.

2.

Decision Making Under No Generation Conditions:

○

The GPC method uses the predictive capabilities of MPC to forecast future load requirements and ESS status. If the wind and PV generation are predicted to remain zero for an extended period, the GPC will optimise the discharging process to sustain the load for as long as possible, considering the constraints such as minimum SOC levels to prevent deep discharging and potential damage to the ESS.

○

The optimisation will balance the discharging power to ensure that the ESS can cover the load demand over the required time horizon. However, if the SOC is low and cannot sufficiently cover the load, the GPC might initiate a strategy to reduce demand (load shedding) or signal the need for external grid support, if available.

2.6. A Tutorial Example

To provide numerical results for the tutorial example, we can calculate the optimal control actions (charging and discharging) of the ESS over the prediction horizon of one-time step $\left(k\right)$ using the GPC method.

Given Data and Parameters:

1.

Initial conditions:

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Initial State of Charge (SOC) of ESS: SOC (1): 50%

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Maximum charging power: $P_{char}^{max}=100 \mathrm{kW}$

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Maximum discharging power: $P_{dis}^{max}=100 \mathrm{kW}$

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Initial charging efficiency: ${\eta }_{char}\left(1\right)=\%95$

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Initial discharging efficiency: ${\eta }_{dis}\left(1\right)=\%90$

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Weighting factors: $w_1=0.5$, $w_2=0.3,$ $w_3=0.2$

2.

Forecasted load and generation:

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At $k$:

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$P_{load}\left(k\right)=50 \mathrm{kW}$, $P_{PV}\left(k\right)=0 \mathrm{kW}$ and $P_{WT}\left(k\right)=10 \mathrm{kW}$

 

Step-by-Step Optimisation:

 

At time step k:

1.

Power balance: $P_{gen}\left(k\right)=P_{PV}\left(k\right)+P_{WT}\left(k\right)+P_{dis}\left(k\right)-P_{char}\left(k\right)=0+10+P_{dis}\left(k\right)-P_{char}\left(k\right)$

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The required power balance to meet the load is: $P_{gen}\left(k\right)=P_{load}\left(k\right)=50 \mathrm{kW}$

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Thus,

$$P_{dis}\left(k\right)-P_{char}\left(k\right)=40$$

2.

Determine Optimal Charging/Discharging Strategy: To meet the load, we need a net discharge of 40 kW. Since the ESS cannot charge and discharge simultaneously, we have two cases:

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Case 1: Discharging Only—If the ESS is discharging to cover the deficit,

$$P_{char}\left(k\right)=0,\hspace{1em}\hspace{1em}\hspace{1em}{P}_{dis}\left(k\right)=40 \mathrm{kW}$$

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Case 2: Charging Only—Since we need to meet the load, charging only would not be feasible in this scenario, so

$$P_{char}\left(k\right)=0$$

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Thus, the optimal strategy here is to discharge 40 kW from the ESS.

3.

Update SOC After Discharging: The SOC of the ESS after discharging at time step $k=1$ is calculated using the following formula:

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From Equation (14), $SOC\left(k+1\right)=50\%+\left(0.95 \times 0\right)-(0.90 \times 40)$

$$SOC\left(k+1\right)=14\%$$

After discharging 40 kW to meet the load at time step $k=1$, the SOC of the ESS decreases from 50% to 14%. The ESS can support the load at this time step by discharging, but its SOC drops significantly, leaving limited capacity for further discharging in subsequent time steps. This result highlights the need for careful management of the ESS to prevent the SOC from dropping too low and ensure reliable MG operation.

3. Results

3.1. The Case Study Description

To provide a comprehensive understanding of the MG system and the context in which the GPC method is applied, this section presents the parameters and data used in the case study, including the solar irradiation and wind speed curves, capacities of DERs, load profiles, efficiencies, ESS capacity, and price signals. These parameters as shown in

Table 1. Solar irradiation, wind speed, and load demand profile for the building.

Time (Hours)	Solar Irradiation (W/m2)	Wind Speed (m/s)	Load Demand (kW)
01:00	0	12	50
02:00	0	11	45
03:00	0	10	43
04:00	0	9	35
05:00	50	8	30
06:00	100	6	40
07:00	200	5	54
08:00	400	14	60
09:00	600	12	74
10:00	800	10	80
11:00	1000	10	90
12:00	1100	7	105
13:00	1000	7	97
14:00	800	8	98
15:00	600	9	99
16:00	400	7	87
17:00	200	11	102
18:00	100	13	101
19:00	50	15	105
20:00	0	14	98
21:00	0	13	99
22:00	0	7	96
23:00	0	6	94
24:00	0	7	90

and

Table 2. Some parameters for the study.

Parameters	Values	Parameters	Values
ESS capacity	150 kWh	Daytime price (7 AM–7 PM)	USD 0.20 per kWh
PV panels	100 kW	Nighttime price (7 PM–7 AM)	USD 0.10 per kWh
WTs	90 kW	${SOC}_{max}$	0.9 (90%)
$n_{char}\left(1\right)$	0.95	${SOC}_{min}$	0.15 (15%)
$n_{dis}\left(1\right)$	0.90	$SOC\left(1\right)$	0.5 (50%)
$P_{i,min}$	0 kW	$R$	0.5 ohms
$P_{i,max}$	110 kW	$N_P$	24 h
$w_1$	1.0	$N_C$	5 h
$w_2$	0.6	Mutation rate	0.1
$w_3$	0.4	Emission factor	0.22499 kg CO2e per kWh
$A_{PV}$	200 m2	${\beta }_{PV}$	−0.0045 (°C−1)
${\eta }_{PV}$	0.18 (18%)	$A_{WT}$	90 m2
$P_{rated}$	50 kW	$v_{cut\_in}$	3 m/s
$v_{cut\_out}$	25 m/s	$v_{rated}$	12 m/s

are crucial for understanding the simulation results and the effectiveness of the proposed GPC method in optimising the MG’s performance. η_char(1) and η_dis(1) are the initial values of charging and discharging efficiency in Table 2.

MPC, a core component of the GPC method, is particularly adept at handling non-linearities and dynamic changes in the system. MPC’s predictive capabilities allow it to anticipate future states and adjust control actions, accordingly, ensuring optimal performance. This is especially beneficial in managing the complexities of MGs, where demand and generation can be highly variable. MPC’s ability to incorporate constraints directly into the optimisation process ensures that the system operates within safe and efficient limits, enhancing both reliability and sustainability. It is worthy to note that in GPC, the combination of different genetic operators (mutation and crossover) with selection strategies (random selection and elitism) is critical for optimising the performance of the algorithm. This study considers four distinct scenarios by combining these operators and strategies to comprehensively evaluate their impact on the optimisation process.

3.2. Mutation–Random Selection

The integration of MPC within the GA offers several benefits, particularly in handling non-linearity and improving overall performance. MPC enhances GPC by predicting future system behaviours and optimising control actions accordingly. The comparison of three methods—MPC, the GA, and the proposed GPC, which combines the strengths of MPC and GAs—highlights the superior performance of GPC in both cost efficiency and environmental impact. Over the course of a day, GPC achieved the lowest total cost of USD 110.83 (Figure 3a) and the lowest total emissions of 52.83 kg CO2e (Figure 3b), significantly outperforming both the MPC and GA methods. The key advantage of MPC lies in its capability to handle system non-linearity and make real-time adjustments, ensuring robust performance under varying conditions. However, while MPC is effective in managing non-linearities, it often results in higher costs and emissions compared to GPC. By integrating the optimisation strengths of GAs with MPC’s real-time adaptability, GPC not only maintains lower costs and emissions consistently throughout the day but also leverages MPC’s robustness in handling complex system dynamics. This combination makes GPC a more effective and sustainable approach for optimising energy usage, providing significant cost savings and reducing environmental impact.

3.3. Mutation–Elitism

In Figure 4, the comparison of MPC, the GA, and the GPC method, which synergises MPC’s capability to handle system non-linearity with GA’s optimisation strengths, clearly demonstrates the superior performance of GPC. The GPC method achieves the lowest total cost of USD 113.94 (Figure 4a) and the lowest total emissions of 52.83 kg CO2e over a day (Figure 4b), outperforming both the MPC method, which has a cost of USD 127.23 and emissions of 71.46 kg CO2e, and the GA method, which results in a cost of USD 144.64 and emissions of 73.73 kg CO2e. While MPC effectively manages non-linearities, its cost and emissions are higher compared to GPC. GA, despite its optimisation capabilities, shows the highest cost and emissions. Therefore, GPC stands out as the most effective and sustainable approach, offering significant cost savings and reduced environmental impact by integrating the best aspects of both MPC and GA.

3.4. Crossover–Random Selection

In Figure 5, the GPC method achieves the lowest total cost of USD 113.94 and the lowest total emissions of 52.83 kg CO2e over a day, outperforming the MPC method, which incurs a cost of USD 136.06 and emissions of 67.85 kg CO2e, as well as the GA method, with a cost of USD 131.90 and emissions of 69.10 kg CO2e. While MPC effectively manages non-linearities, its cost and emissions are higher compared to GPC. The GA, despite its optimisation capabilities, shows higher costs and emissions than GPC. Therefore, GPC stands out as the most effective and sustainable approach, offering significant cost savings and a reduced environmental impact by integrating the best aspects of both MPC and GAs.

3.5. Crossover–Elitism

In Figure 6a, in the Crossover–Elitism approach, GPC achieves the lowest total cost of USD 113.94 and the lowest total emissions of 52.83 kg CO2e over a day. This performance is significantly better than that of MPC, which has a cost of USD 122.76 and emissions of 69.53 kg CO2e, and the GA, which incurs a cost of USD 127.80 and emissions of 69.71 kg CO2e (Figure 6b). While MPC effectively manages system non-linearity, it results in higher costs and emissions relative to GPC. The GA, despite its strong optimisation abilities, performs worse than GPC in both cost and emissions. Thus, GPC stands out as the most effective and sustainable approach, combining the robustness of MPC with the optimisation efficiency of GAs, leading to significant cost savings and reduced environmental impact.

4. Discussion

The integration of MPC within GPC showcases significant advantages over the traditional GA across different scenarios. The analysis of the four sets of results highlights these benefits in terms of cost efficiency, emissions reduction, power generation, and overall optimisation performance. The results of the comparison between MPC, GA, and the proposed GPC highlight the distinct advantages and performance metrics of each method in the context of optimising energy usage and reducing environmental impact.

4.1. Performance Analysis

Cost Efficiency: The GPC method consistently demonstrates the lowest total cost for the day across various scenarios, including Mutation–Elitism and Crossover–Elitism. Specifically, the GPC method achieves a lower total cost of USD 113.94 compared to the costs of both MPC and GA in the Crossover–Elitism scenario, specifically highlighting that GPC’s cost is lower than MPC’s USD 122.76 and GA’s USD 127.80. This cost efficiency can be attributed to GPC’s ability to effectively combine the real-time adaptability and non-linearity handling of MPC with the robust optimisation capabilities of the GA. This synergy allows GPC to minimise costs by making more informed and precise adjustments throughout the day.

Environmental Impact: In terms of emissions, GPC again outperforms both MPC and the GA. The GPC method achieves the lowest total emissions of 52.83 kg CO2e, significantly lower than MPC’s 69.53 kg CO2e and the GA’s 69.71 kg CO2e in the Crossover–Elitism scenario. The reduction in emissions is a crucial advantage, as it underscores the potential of GPC to contribute to sustainability goals and reduce the carbon footprint of energy systems. The lower emissions are a direct result of the GPC’s optimised control strategy, which ensures efficient energy usage and reduces wasteful emissions. The realisation of CO₂ emissions in the context of this paper is managed by optimising the EMSs for the MG using the GPC method. The GPC method aims to reduce emissions by optimising the balance between power generation, load demands, and the use of ESS. The emissions are quantified as the total emissions produced by the MG, and this is a key metric used to evaluate the effectiveness of the GPC method. By combining MPC and a GA, the GPC method enhances the capability to reduce CO₂ emissions through more efficient resource allocation and improved handling of non-linearities and non-convexities in the system model.

4.2. Methodological Insights

MPC: MPC’s strength lies in its ability to handle non-linearities and make real-time adjustments based on system dynamics. This capability ensures robust performance under varying conditions. However, MPC’s approach, while effective in managing non-linearities, often results in higher operational costs and emissions compared to GPC. The peaks in cost and emissions observed in the MPC results suggest that while it adapts well to changes, it may not always find the most cost-effective or environmentally friendly solution.

GA: The GA is known for its powerful optimisation capabilities, which enable it to explore a wide solution space and identify optimal strategies. However, the GA’s higher variability in cost and emissions indicates that it may lack the real-time adaptability required to handle dynamic changes as effectively as MPC. The results show that the GA, although efficient in optimisation, tends to incur higher costs and emissions compared to GPC, particularly when not combined with a robust real-time control mechanism.

GPC: The GPC method, which integrates the strengths of both MPC and the GA, emerges as the superior approach. GPC leverages MPC’s ability to handle non-linearities and make real-time adjustments while utilising GA’s optimisation strength to continuously refine control strategies. This combination allows GPC to maintain lower costs and emissions consistently. The fitness graphs further support this, showing that GPC achieves higher and more stable fitness levels throughout the iterations, indicating a more effective and sustained optimisation process.

4.3. Limitations and Challenges

Although the GPC method offers a number of advantages, important improvements over traditional approaches, there are a number of limitations and challenges that need to be addressed for practical implementation and wide diffusion.

Computational Complexity: While effective in optimising MG operations, the integration of GAs and MPC in the GPC method results in high computational complexity. In that respect, the intrinsic interactivity of the GA adds to the predictive capabilities brought about by MPC, which requires high computational effort. This might be more problematic in real-time applications; specifically, where fast decision making is necessary. This computational burden may limit applicability in smaller MGs or scenarios where advanced computation infrastructures are unavailable. While the GPC method offers substantial benefits in terms of efficiency, sustainability, and cost-effectiveness, addressing the aforementioned limitations and challenges is crucial for its successful implementation. Future research should focus on developing solutions to mitigate these challenges, enhancing the practicality and robustness of the GPC method in diverse MG settings.

5. Conclusions

The comparative analysis of MPC, GAs, and the proposed GPC method underscores the superior performance and advantages of GPC in optimising energy usage. GPC, which synergises MPC’s adept handling of non-linearity with the GA’s powerful optimisation capabilities, consistently achieves the lowest total cost and emissions across different scenarios, including Mutation–Elitism and Crossover–Elitism. The GPC demonstrated the lowest total cost of USD 113.94 and the lowest total emissions of 52.83 kg CO2e, significantly outperforming both MPC and GA. This performance highlights GPC’s ability to maintain economic efficiency and environmental sustainability, making it a more attractive option for energy management. While MPC effectively manages system non-linearities and adapts to real-time changes, it incurs higher costs and emissions compared to GPC. MPC’s robust performance under dynamic conditions is valuable, but it does not achieve the same level of cost-effectiveness or environmental efficiency as GPC. The GA excels in exploring a wide solution space and identifying optimal strategies, but its higher cost and emission levels indicate that it lacks the real-time adaptability necessary for dynamic energy management. When used alone, the GA does not perform as efficiently as GPC. The GPC leverages the best features of both MPC and the GA, resulting in superior performance in terms of cost and emissions. By integrating MPC’s real-time control with the GA’s optimisation, GPC provides a balanced and highly effective approach to energy management. Implementing GPC in real-world energy systems can lead to substantial cost savings and significant reductions in emissions, supporting both economic and environmental goals. Its robust performance across various scenarios suggests that GPC is well-suited for diverse applications, including renewable energy integration and industrial energy management. The GPC method stands out as the most effective and sustainable approach for optimising energy usage, combining the strengths of MPC and the GA to achieve significant cost savings and a reduced environmental impact. Its adoption can play a crucial role in addressing modern energy management challenges and contributing to global sustainability efforts.