Multi-Objective Energy Optimization with Load and Distributed Energy Source Scheduling in the Smart Power Grid

Abstract

Multi-objective energy optimization is indispensable for energy balancing and reliable operation of smart power grid (SPG). Nonetheless, multi-objective optimization is challenging due to uncertainty and multi-conflicting parameters at both the generation and demand sides. Thus, opting for a model that can solve load and distributed energy source scheduling problems is necessary. This work presents a model for operation cost and pollution emission optimization with renewable generation in the SPG. Solar photovoltaic and wind are renewable energy which have a fluctuating and uncertain nature. The proposed system uses the probability density function (PDF) to address uncertainty of renewable generation. The developed model is based on a multi-objective wind-driven optimization (MOWDO) algorithm to solve a multi-objective energy optimization problem. To validate the performance of the proposed model a multi-objective particle swarm optimization (MOPSO) algorithm is used as a benchmark model. Findings reveal that MOWDO minimizes the operational cost and pollution emission by 11.91% and 6.12%, respectively. The findings demonstrate that the developed model outperforms the comparative models in accomplishing the desired goals.

1. Introduction

Around the globe, pollution emissions and global warming are major concerns to the environment and sustainability causing climate change [1,2]. The demand response (DR) and renewable energy sources (RES) emerged to control pollution emissions from power system networks [3,4]. Thus, power sector regulatory bodies are attempting to proliferate and penetrate RESs, such as wind, solar, fuel cell, etc., to meet growing electricity demand with low pollution emissions [5,6]. With the emergence of the smart power grid (SPG) [7], the utility can accommodate RESs and initiate DR programs, contribute to reducing pollution emissions, operation cost minimization, power loss reduction, and improved reliability [8]. The utility involves consumers in DR programs to engage in the electricity market with the aim of addressing energy optimization issues, including contingency problems, RES output forecast errors, pollution emissions, operational cost, and distributed grounding [9]. The DR programs encourage consumers to adapt their energy consumption profile, including shifting load, and curtailing load during high-price hours, to minimize pollution emissions and operation costs [10,11].

Numerous studies have been undertaken regarding energy optimization in the SPG, covering various aspects and viewpoints such as dynamic pricing policies [12], recognition methods [13], DFT phasor estimators [14], control algorithms [15], etc. However, RES modelling, scheduling, and optimization are necessary before addressing the above aspects and viewpoints in the literature. For instance, authors developed a model to cater for the erratic and uncertain behaviour of RESs, such as the wind [16]. Likewise, PDF and CDF are studied in [17,18] for the uncertainty modelling perspectives of RESs such as wind and solar. Similarly, distributed generation and microgrid equal power sharing and control are addressed in [19,20]. Moreover, consumers can reduce their bills by participating in DR programs. DR programs are implemented in [21] for SPG cost and carbon emission reduction. The DR motivate domestic, industrial, and commercial consumers to schedule their load demand and optimize energy via incentive-based payments for energy cost minimization. Optimization methods such as Lyapunov optimization and distributed algorithm are used to optimize the cost of microgrids with the high proliferation of RESs [22,23]. A multi-objective optimization model for hybrid systems haseen developed using multi-criterion decisions for environmental and economic aspect optimization [24]. Similarly, the NSGA-II model was developed considering rural home’s flexible load scheduling for financial benefit and user comfort maximization [25]. The authors of [26] introduced shuffle frog leaping and teaching/learning-based optimization algorithms for different prospectives of optimization such as energy utilization, energy cost reduction, and creating energy balance between utility and consumers. A robust optimization via Monte Carlo simulations (MCS) was introduced in [27] to accommodate the high penetration of RESs using storage batteries. The primary goal of this research included reducing energy consumption, minimizing PAR, optimizing electricity expenses at peak demand, and minimizing energy wastage through various techniques: real-time pricing (RTP), time of use (TOU), optimum load management (OLM), decision support tool (DST) and critical peak pricing (CPP). A multi-objective optimization model was introduced in [28] for microgrid carbon capturing. Likewise, a multi-objective approach was developed for intelligent decision making of PMU placement in the grid [29].

The authors of [30] developed a residential sector energy optimization model to control consumer load behaviour, balance supplied power, reduce electricity bills, and improve reliability. A distributed algorithm was introduced in [31,32] to solve economic dispatch problems via a multi-objective approach. The authors of [33] introduced a model considering electric vehicle (EV) scheduling for the charge and discharge process using an adaptive simulated annealing particle swarm optimization algorithm to provide financial relief and reduce emissions. A mixed integer non-linear programming (MINLP) model was proposed to minimize pollution emissions and operational costs in a microgrid [34]. The obtained results of the MINLP were compared with existing algorithms for validation. Optimizing energy purchasing and selling with storage batteries was discussed in [35]. An optimization model for DSM in a SPG was studied, optimizing the operation of shiftable/adjustable electrical and thermal load considering RESs, battery storage, etc., to minimize cost and shave peaks [36]. A stochastic multi-objective model considering RESs and EVs was presented in [37] to minimize operating costs and voltage deviation. A fuzzy control mechanism was introduced in [38,39,40] to balance microgrid and power grid voltage and current sharing. Electricity theft detection and residential load decomposition was addressed in [41,42]. The authors of [43] developed a chaotic hybrid sine cosine pattern search algorithm to solve the optimization problem in power system stabilizers. The authors of [44] addressed the golden search optimization algorithm for numerical function optimization. The developed model scheduled load using stochastic techniques under DR. Findings illustrated that the developed model outperformed with/without DR in minimizing cost and peak energy consumption. GA and MOGA were adopted in [45] for the single/multi-objective optimization of environmental and economic aspects. Similarly, multi-objective stochastic algorithms considering DR, energy storage, and RESs for retailer energy cost minimization and clean energy utilization maximization were discussed in [46]. In the work [47], an optimization model was implemented, modelling the erratic behaviour/uncertainty of solar and wind energy. In [48], MOPSO was developed to minimize pollution emissions and operational costs in the SPG. Both operating expenses and emissions were decreased at the same time. The proposed model successfully optimized the operating expenses and emissions compared to existing models. The model developed in this study is compared with previously published models, as listed in

Table 1. Comparison of the developed model with existing models in the discussed literature.

Refs.	Objectives	Techniques	Loads	Optimization	Uncertainty
[16]	Operation cost, pollution emission, and load curtailment cost minimization	MOWDO and MOGA	Responsive and non-responsive	Multi-objective	PDF
[22,23]	Energy cost minimization	TLBO, SFL, EDGE	Residential load	Single-objective	–
[49]	Environmental, economic, and optimal shifting	Augmented $ϵ$-constraint	Thermal and electrical shiftable load	Multi-objective	PDF
[50]	Operation expenses and environmental pollution minimization	MOWDO and MOGA	Responsive and non-responsive	Multi-objective	PDF
[27]	Purchase cost and user’s dissatisfaction minimization	MOGA	Domestic load	Multi-objective	–
[33]	Operating cost and environmental governance cost minimization	Adaptive simulated annealing particle swarm optimization algorithm	Residential, commercial, and industrial	Multi-objective	PDF
[34]	Economic and environmental cost minimization	MINLP	Residential	Multi-objective	–
[51]	Operation costs, emission pollution, loss of energy supply probability	$ϵ$-constraint	Thermal and electrical load	Multi-objective	–
[35]	Loss of power supply probability and energy cost minimization, and maximizing renewable generation availability	MOPSO	Village load	Multi-objective	–
[36]	Peak shaving and cost curtailment	MINLP	Electrical and thermal load	Single-objective	–
[37]	Operating cost and voltage deviation minimization	$ϵ$-constraint	IEEE 34-bus test system	Multi-objective	Roulette wheel
[52]	Emission, cost, LOLE, and deviation	$ϵ$-constraint	Thermal and electrical load	Multi-objective	–
[45]	Least cost and least environmental footprint	GA and MOGA	Residential	Single/Multi-objective	–
[46]	Retailers energy cost minimization and clean energy utilization maximization	Stochastic algorithm	Electricity retailers	Multi-objective	–
[53]	Operation costs, emission pollution and customer satisfaction	Shuffled frog leaping algorithm	Thermal and electrical load	Multi-objective	Log normal distribution function
[48]	Economical and environmental situation optimization	MOPSO	Domestic, industrial, and commercial	Multi-objective	PDF
[54]	Operation cost and pollution emission	MOGA and MOPSO	Domestic, industrial, and commercial	Multi-objective	PDF
[55]	Environment and economic, and load curtailment cost	$ϵ$-constraint	Thermal and electrical load	Multi-objective	–
This work	Operational cost, clean energy utilization, pollution emission, uncertainty handling of generation and load	MOWDO and MOPSO	Domestic, industrial, and commercial	Multi-objective	MCS and PDF

.

The above-discussed works comprehensively analyse the existing literature, investigating different aspects, perspectives, and methodologies, and develop an in-depth understanding of achieving objectives including operating cost, pollution emission, etc. Nevertheless, how the multi-objective optimization problem is addressed differs from the existing literature in terms of methodology, techniques, scenario, and validation. This work develops a multi-objective optimization model for scheduling load and distributed energy sources in the SPG to minimize operational costs and pollution emissions. Furthermore, this work employs the multi-objective wind-driven optimization (MOWDO) method, incorporating the Pareto criterion and non-linear sorting based on a fuzzy mechanism to solve the multi-objective optimization problem. The novelty and technical advancements of this research are summarized below.

• The PDF is used to address the uncertainty of wind and PV renewable energy to participate effectively in energy optimization.
• Optimization model using the MOWDO technique is devised with a fuzzy approach to solve multi-objective optimization issues of industrial, residential, and commercial consumers.
• The developed model’s results are better than the benchmark model in terms of both operational expenses and pollution emissions.

This paper is structured as follows: Section 2 extensively describes the developed system. The created multi-objective optimization algorithm MOWDO is presented in Section 3. Finally, the simulation results are discussed in Section 4, followed by the conclusion in Section 5.

2. Proposed System Model

Due to the growing need for energy, the world is evolving from traditional energy sources to RESs and combined RESs, forming distributed generation (DG). The SPG accommodates DG and facilitates bidirectional communication between the energy source and the load for energy optimization. This two-way/bidirectional communication is possible with the advanced metering infrastructure (AMI) of the SPG that stores and processes data from consumers and sources through smart meters and sends it back to the service provider. This real-time information exchange allows users to balance power generation and load consumption via DR incentives. A microgrid can either work in solitary mode or be mixed with renewable energy in harmony to provide energy. The developed system model is shown in Figure 1.

2.1. Consumers

The developed model considers three types of consumers:

• Domestic;
• Industrial;
• Commercial.

where $\left(\mathrm{RC}\right)$:$\mathrm{RP}(\mathrm{r},\mathrm{t})=\mathrm{RC}(\mathrm{r},\mathrm{t})$, $\left(IC\right)$:$IP(i,t)=IC(i,t)$, and $\mathrm{CP}(\mathrm{c},\mathrm{t})=\mathrm{CC}(\mathrm{c},\mathrm{t})$ are for the above consumers, respectively. There are different load categories for consumers, shiftable and non-shiftable loads. Non-shiftable loads are used around the clock and cannot be turned off or shifted. These types of loads include fans, bulbs, refrigerators, etc. A domestic consumer can change home appliances, such as washing machines, iron, and air conditioner, to avoid high prices. The utility service provider provides cost incentives for consumers to shift load from on-peak hours (bearing high cost) to off-peak hours where the tariff rate is relatively lower. This helps in peak shaving and reduces power demand over peak hours.

2.2. RES Uncertainty Modelling

This study considers two types of RESs, namely solar and wind, that exhibit uncertainty, stochasticity, and intermittency due to their dependence on environmental factors. To smooth-out RES fluctuations, batteries are used with RES [56,57]. In contrast, to model the uncertainty of RESs, probabilistic models are developed because wind speed and solar irradiance are stochastic variables that use meteorological data to estimate their generation potential. A detailed discussion on wind and solar energy uncertainty modelling is as follows.

2.2.1. Wind Energy Modelling

The wind is a renewable source used to run wind turbines (WTs) to convert wind speed into electrical energy [58,59,60]. The energy generated from WTs depends upon the turbine’s size and shape, wind availability, and wind speed quotient [61]. The bigger the turbine blades, the more energy is produced from the turbine [62]. Likewise, WTs in higher wind speeds generates more energy. Due to the stochastic nature of wind speed, we propose a probabilistic model for estimating the amount of energy it can produce. To model the behaviour of wind speed, the PDF is presented in [63], where ${\alpha }_{\omega }$, ${\beta }_w$, etc. are scale/shape parameter, which assumed to be 2. Wind speed is modelled using PDF, depicted in Figure 2.

Let $v_m$ be the mean wind speed, then the scale parameter of a particular site is given below [54].

$$F_v\left(v_{\mathrm{wind}}\right)=1-exp\left(-{\left(\frac{v_{wind}}{{\alpha }_{\omega }}\right)}^2\right)$$

(1)

$$f_v\left(v_{wind}\right)=\frac{2}{{\alpha }_{\omega }^2}{v}_{wind}exp\left(-{\left(\frac{v_{wind}}{{\alpha }_w}\right)}^2\right)$$

(2)

Substituting $a_w$ as an average wind speed in PDF, the Rayleigh distribution function for wind energy of the above equations is obtained as follows [54].

$$F_v\left(v_{\mathrm{wind}}\right)=\frac{\pi }{2}\frac{v_{wind}}{v_{m^2}}exp\left(-\left(\frac{\pi }{4}\right)\left(\frac{v_{wind}}{v_{m^2}}\right)\right)$$

(3)

$$f_v\left(v_{wind}\right)=1-exp\left(-\left(\frac{\pi }{4}\right){\left(\frac{v_{wind}}{v_m}\right)}^2\right)$$

(4)

The output characteristic of wind energy can be obtained using Equation (5) adopted from [54,64] and modelled below.

$$p_w\left(v_{wind}\right)=\left\{\begin{array}{cc}0\hfill & v_{wind}<v_{ci}\hfill \\ P_R=\frac{\left(v_{\mathrm{wind}}-v_{ci}\right)}{\left(v_r-v_{ci}\right)}\hfill & v_{ci}\le v_{wind}<v_r\hfill \\ P_R\hfill & v_r\le v_{wind}<v_{co}\hfill \\ 0\hfill & v_{\mathrm{wind}}\ge v_{co}\hfill \end{array}\right.$$

(5)

where $v_r,v_{ci}$ and $v_{\mathrm{wind}}$ are the rated, cut in, and factual wind speed, respectively. The $P_R$ is the rated power of WTs. The parameters used for WTs used in this proposed model have $v_{ci}=3.5\frac{\mathrm{m}}{\mathrm{s}};v_{co}=18\frac{\mathrm{m}}{\mathrm{s}};v_r=$$17.5\frac{m}{s}$. Figure 3 shows wind speed profile used in wind energy generation.

2.2.2. Solar Photovoltaic Modelling

Solar is a form of renewable energy that produces energy from solar radiation and temperature via the PV effect [65]. The solar-generated energy depends on the intensity of the sunlight: solar radiation and temperature [66,67]. The solar irradiance uncertainty is modelled via PDF functions in Equation (6) as [54].

$$f_B\left(si\right)=\left\{\begin{array}{c}\frac{r(\alpha +\beta )}{r\left(\alpha \right)r\left(\beta \right)}s{i}^{\alpha -1}{(1-si)}^{\beta -1}0\le si\le 1,\alpha \ge 0\hfill \\ 0\mathrm{otherwise}\hfill \end{array}\right.$$

(6)

$$F_B={\int }_0^{si}\frac{r(\alpha +\beta )}{r\left(\alpha \right)r\left(\beta \right)}s{i}^{\alpha -1}{(1-si)}^{\beta -1}dsi\beta \ge$$

(7)

where $si$ denotes solar radiations. The $\alpha$/$\beta$ are parameters, where the average solar irradiance and standard deviation from the PDF are calculated and used below [54].

$$\alpha =\mu \left(\frac{\mu (1+\mu )}{{\sigma }^2}-1\right)\beta =(1-\mu )\left(\frac{\mu (1+\mu )}{{\sigma }^2}-1\right)$$

(8)

Solar output power is obtained using Equation (9) [68,69,70].

$$p_h\left(si\right)=A_c\times \eta \times si$$

(9)

where $p_h\left(si\right)$ denotes the PV output power obtained from the solar irradiance $si$, $\eta$ represents PV module efficiency, and $A_c$ PV module is surface area. The uncertainty of the PV output power with PDF is modelled in Equation (10).

$$f_{P_{pv}}\left(P_{pv}\right)=\left\{\begin{array}{c}\frac{r\left(\alpha +\beta \right)}{r\left(\alpha \right)r\left(\beta \right)}{\left(p_h\left(si\right)\right)}^{\alpha -1}{\left(1-p_h\left(si\right)\right)}^{\beta -1}\hfill \\ if{P}_{pv}\in \left[0,p_h\left(si\right)\right]\hfill \\ 0otherwise\hfill \end{array}\right.$$

(10)

The solar PV output power is depicted in Figure 4. The bids and emissions coefficient of PV are listed in

Table 2. Bids and emissions coefficient of the sources considered in the developed model.

Source	Bid	Start/Shut Cost	${\mathbf{CO}}_2$	${\mathbf{SO}}_2$	${\mathbf{NO}}_{\mathit{x}}$	${\mathit{P}}_{\mathbf{min}}$	${\mathit{P}}_{\mathbf{mx}}$
WT	1.073	0	0	0	0	0	15
PV	2.584	0	0	0	0	0	25
Bat	0.38	0	10	0.0002	0.001	−30	−30
MT	0.457	0.96	720	0.0036	0.1	6	30
Grid	-	0	950	0.5	2.1	−30	30

.

2.3. Objective Functions

The developed model balances demand and generation to optimize pollution emission and operational cost as objectives considering DGs and demands like residential, industrial, and commercial. Objectives are defined as follows.

2.3.1. Operational Cost

The first objective function is operational cost optimization. For all the sources participating in energy production, the cost function is subdivided into energy reserve cost and exchange with the utility grid as well as the cost of starting up and shutting down energy production units. Our energy production unit, which includes diesel generators, fuel cells, microturbines, wind, and solar energy systems, operates in coordination to supply energy consistently and meet energy demands. The cost function is modelled as follows [71].

$$minf1\left(x\right)=\sum _{t=1}^{T}S{T}_{DG}\left(t\right)+cos{t}_{DG}\left(t\right)+cos{t}_{\mathrm{Gd}}\left(t\right)+cos{t}_s\left(t\right)$$

(11)

$$S{T}_{DG}\left(t\right)=\sum _{i=1}^{Ng}{S}_{DGi}\left|U_i\left(t\right)-U_i(t-1)\right|$$

(12)

$$\begin{array}{c}cos{t}_{\mathrm{Gd}}\left(t\right)=U_{By}\left(t\right)P_{\mathrm{Gd}-\mathrm{By}}\left(t\right)B_{\mathrm{Gd}-\mathrm{By}}\left(t\right)\hfill \\ -U_{\mathrm{sell}}\left(t\right)P_{\mathrm{Gd}-\mathrm{sell}}\left(t\right)B_{\mathrm{Gd}-\mathrm{sell}}\left(t\right)\hfill \end{array}$$

(13)

$$Cos{t}_{DG\left(t\right)}=\sum _{i=1}^{Ng}{U}_i\left(t\right)P_{DGi}\left(t\right)B_{DGi}\left(t\right)$$

(14)

$$cos{t}_s\left(t\right)=\sum _{j=1}^{Ns}\left|\begin{array}{c}{u}_j\left(t\right)\times P_{sj}\left(t\right)\times B_{sj}\left(t\right)+\hfill \\ \left(S_{sj}\left(t\right)\times u_j\left(t\right)-u_j(t-1)\right)\hfill \end{array}\right|$$

(15)

$S_{DGi}\left(t\right)$ and $S_{sj}\left(t\right)$ denote the startup or shutdown costs for the turbine and storage units, respectively, $P_{\mathrm{Gd}-\mathrm{By}}\left(t\right)$ indicates the real power purchased from the utility, and $P_{\mathrm{Gd}-\mathrm{sell}}\left(t\right)$ refers to the real power sold to the utility. The bids trade (purchase and sell) with the power company are $B_{\mathrm{Gd}-\mathrm{By}}\left(t\right)$ and $B_{\mathrm{Gd}-\mathrm{sell}}\left(t\right)$, respectively, are listed in Table 2. $P_{DGi}\left(t\right)$ and $P_{sj}\left(t\right)$ are the actual output power of the generators and battery storage, respectively. Likewise, $B_{sj}\left(t\right)$ and $B_{DGi}\left(t\right)$ are the bids of the storage devices and DGs, respectively.

2.3.2. Pollution Emissions

Pollution emissions comprise the quantity of pollution emitted from DGs, generators, and utility grids, mathematically modelled in the pollution function below [71].

$$minf2\left(x\right)=\left\{\begin{array}{c}\sum _{i=1}^{Ng}\left[U_i\left(t\right)P_{DGi}\left(t\right)E_{DGi}\left(t\right)\right]+\dots \hfill \\ \sum _{j=1}^{Ns}\left[U_j\left(t\right)P_{sj}\left(t\right)E_{sj}\left(t\right)+P_{Gd}\left(t\right)E_{Gd}\left(t\right)\right]\hfill \end{array}\right.$$

(16)

where $E_{si}\left(t\right)$, $E_{DGi}\left(t\right)$, and $E_{Gd}\left(t\right)$ are pollution emissions measured in kgMWh${}^{-1}$ for the generator units, DGs, and utility, respectively.

2.4. Constraints

The developed model ensures optimal energy optimization of the SPG by considering the following constraints.

2.4.1. Energy Balancing Constraints

The power generation (DGs, generator, power grid, etc.) must equal the net load demand to ensure energy balance. Energy balancing constraints are defined as follows [71].

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \sum _{i=1}^{Ng}}{P}_{DGi}\left(t\right)+{\displaystyle \sum _{j=1}^{Ns}}{P}_{sj}\left(t\right)+U_{\mathrm{By}}\left(t\right)P_{\mathrm{By}}\left(t\right)P_{\mathrm{Gd}-\mathrm{By}}\left(t\right)\hfill \\ ={\displaystyle \sum _{L=1}^{N_L}}{P}_{\mathrm{Demand}}\left(t\right)+U_{\mathrm{sell}}\left(t\right)P_{\mathrm{Gd}-\mathrm{sell}}\left(t\right)\hfill \end{array}\end{array}$$

(17)

where $N_L$ is demand levels number and $P_{Demand}\left(t\right)$ represents power demand to be met by the SPG.

2.4.2. Power Threshold Constraint

The power originating from each source is limited by the source’s minimum and maximum power capacity thresholds, mathematically modelled as follows [71].

$$\begin{array}{c}\hfill P_{DGi,min}\left(t\right)\le P_{DGi}\left(t\right)\le P_{DGi,max}\left(t\right)\\ \hfill P_{sj,min}\left(t\right)\le P_{sj}\left(t\right)\le P_{sj,max}\left(t\right)\\ \hfill p_{\mathrm{Gd},min}\left(t\right)\le P_{Gd}\left(t\right)\le P_{Gd,max}\left(t\right)\end{array}$$

(18)

where $P_{DGi,max}\left(t\right),P_{sj,max}\left(t\right)$, and $P_{\mathrm{Gd},\max }\left(t\right)$ is the maximum real power-generating units of $i^{th}$$DGs,j^{th}$ storage system and the utility, respectively. Furthermore, $P_{DGi,min}\left(t\right),P_{sj,min}\left(t\right)$, and $P_{\mathrm{Gd},\min }\left(t\right)$ are minimum power from the aforementioned sources, respectively.

2.4.3. Battery Constraints

Batteries charging and discharging are limited by their minimum and maximum charging and discharging capacity limits, modelled below.

$$\begin{array}{c}{B}_{ess}\left(t\right)=B_{ess}\left(t-1\right)+{\eta }_{chg}\left(t\right)P_{chg}\left(t\right)\Delta \Delta t\times I_{chg}\hfill \\ -\frac{1}{{\eta }_{dischg}\left(t\right)}\Delta t\times I_{dischg}+I_{chg}\left(t\right)\hfill \\ B_{ess,min}⩽B_{ess}\left(t\right)⩽B_{ess,max}{P}_{chg}\left(t\right)\hfill \\ ⩽P_{chg,max};P_{dischg}\left(t\right)⩽P_{dischg,max}\hfill \end{array}$$

(19)

$B_{ess}$ represents stored energy in battery. Likewise, $P_{chg}$ and $P_{dischg}$ show the charge and discharge power for duration $\delta t$, respectively, while ${\eta }_{chg}$ and ${\eta }_{dischg}$ denote the battery charging/discharging efficiency, respectively. $B_{ess,min}$ and $B_{ess,max}$ show the lower and upper limits for energy storage in the battery, respectively, and $P_{chg,max}$ and $P_{dischg,max}$ denote maximum charge/discharge battery power for a period of $\delta t$, respectively.

The bids and emissions coefficient of sources, such as WTs, PVs, batteries, generators, and power grids, are listed in Table 2.

3. Developed Multi-Objective Optimization Algorithms

The developed model adopted the MOWDO algorithm due to its capability of handling conflicting constraints, equality/inequality constraints, and multi-conflict functions to achieve the desired objectives. The air parcels in the MOWDO algorithm have positions and velocities with five benchmark functions: Kita, Schaffer, Kursawe, ZDT1, and ZDT4, for solving optimization problems, as illustrated in Algorithm 1.

Algorithm 1: MOWDO algorithm for multi-objective energy optimization.

MOWDO uses Pareto-front ranks to determine the global best solution [16].

3.1. Schaffer Benchmark Function

This function’s lower/upper bounds for variables are $[-{10}^3,{10}^3]$, and the optimal solution is bounded as [0, 2]. The Schaffer function is modelled as follows.

$$f_1\left(k\right)=k^2,f_2\left(k\right)={(k-2)}^2$$

(20)

3.2. Kita Function

This function’s variable is bounded as $[0,7]$, and the multi-objective functions are defined as follows.

$$f_1(k_1,k_2)=-{k_1}^2+k_2$$

(21)

and

$$f_2(k_1,k_2)=\frac{k_1}{2}+k_2+1$$

(22)

subject to:

$$\frac{k_1}{6}+k_2\le \frac{13}{2},\frac{k_1}{2}+\frac{15}{2},5k_1+k_2\le 30$$

(23)

The function is designed to mitigate the impact of pressure.

3.3. Kursawe Function

This function’s bounded variables are $[-5,5]$, and their multi-objective function is defined below:

$$f_1\left(k\right)=\sum _{j=1}^{N-1}(-10exp(-0.2\sqrt{{k^2}_i+{k^2}_{j+1}}))$$

(24)

$$f_2\left(k\right)=\sum _{j=1}^N|k_j{|}^{0.8}+5sin({k^3}_j))$$

(25)

3.4. ZDT1 Function

ZDT1 variables are bounded as $[0,1]$, and its multi-objective function is defined below.

$$\begin{array}{c}{f}_1\left(k\right)=k_1\hfill \\ and\hfill \\ f_2\left(k\right)=g\left(k[1-\sqrt{\frac{k_1}{g\left(k\right)}}]\right)\hfill \\ whereg\left(k\right)=1+9(\frac{\sum _{j=2}^N{k}_i}{(N-1)})\hfill \end{array}$$

(26)

3.5. ZDT4 Function

This function’s variables are bounded as $k_j$ = $[-5,5],j=2,3,..,n$ and $k_1$$=[0,1]$, and its multi-objective function is defined below.

$$\begin{array}{c}{f}_1\left(k\right)=k_1\hfill \\ and\hfill \\ f_2\left(x\right)=g\left(k\right)\left[1-\sqrt{\left(\frac{k_1}{g\left(k\right)}\right)}\right]\hfill \\ and,\hfill \\ g\left(k\right)=1+10(N-1)\hfill \\ +{\displaystyle \sum _{i=2}^N}({k^2}_j-10cos\left(4\pi k_j\right))\hfill \end{array}$$

(27)

Step-wise detail of the MOWDO algorithm is illustrated in the flowchart in Figure 5.

4. Simulation Results and Discussion

The developed model was tested via experiments for multi-objective energy optimization considering power generation sources, such as DGs, power grid, generators, etc., and load, such as domestic, industrial, and commercial, as depicted in Figure 6. The price signal of the energy market used by the developed model is displayed in Figure 7.

We have power-generating sources, such as solar, WT, diesel generators, hydroelectric power, electric vehicles, and fuel cells on the generation side. The developed model under these sources optimizes the operational costs and pollution emissions. On the consumer side, we have different kinds of industrial, residential, and commercial loads. The electricity demand for the loads mentioned above is 50, 33 and 10%, respectively. To address the multi-objective optimization problems, the MOWDO technique was utilized. The defined objectives were fed into the benchmark functions of the MOWDO algorithm. The objective function took $\mathrm{x}$ as a variable function, as this problem was set for $24\mathrm{h}$ since the number of variables was set to 24. RESs, such as solar and wind, are subject to high levels of uncertainty. To model this uncertainty of renewable energy, the PDF was used. The predicted behaviour of solar and wind is shown in Figure 8 and Figure 9, respectively. The results are presented in

Table 3. WT and solar PV prediction analysis.

Hours	$\mathbf{WT}$ (kW)	$\mathbf{PV}$ (kW)	Hours	$\mathbf{WT}$ (kW)	$\mathbf{PV}$ (kW)
1	1.7850	0	13	3.9150	23.90
2	1.7850	0	14	2.3700	21.05
3	1.7850	0	15	1.7850	7.875
4	1.7850	0	16	1.3050	4.225
5	1.7850	0	17	1.7850	0.550
6	0.9150	0	18	1.7850	0
7	1.7850	0	19	1.3020	0
8	1.3050	0.200	20	1.7850	0
9	1.7850	3.750	21	1.3005	0
10	3.0900	7.525	22	1.3005	0
11	8.7750	10.45	23	0.9150	0
12	10.410	11.95	24	0.6150	0

. The energy from these sources is subjected to three types of consumers: domestic, commercial, and industrial.

Domestic are loads used in daily households, e.g., fans, light bulbs, TV, room cooler, etc. In comparison, commercial loads include loads used by shops, restaurants, malls, etc., meant for commercial use. In the same way, industrial loads are heavy machinery, and motors used all day. Simulation results were conducted to solve the optimization problem of pollution emissions and operational costs. These objectives are conflicting and tradeoff exists between them. The MOWDO is employed to achieve both objectives simultaneously. Hence, we take minimal and optimal points at Pareto-fronts to draw our results.

The wind power data presented in Table 3 is taken from [72]. The solar energy system is comprises $25\mathrm{kW}$ SOLAREX MSX type, including a solar array of $10\times 2.5\mathrm{kW}$ with $\mathrm{s}=10{\mathrm{m}}^2$ and $\mathrm{h}=18.6\%$. For storage purposes, batteries are used with a minimum and maximum capacity of 10% and 100%, respectively, adopted from [73]. The daily load demand curve for the commercial, residential, and industrial consumer is illustrated in Figure 6, taken from [74] as the daily average load demand curve. Energy used for the span of $24\mathrm{h}$ is shown to be $1695\mathrm{kWh}$ [75]. However, these sources have uncertainty and rigid operation constraints due to dependence on environment [76,77]. It is seen that all sources participate actively to meet the demand in compliance with the smooth functioning of the SPG. The detailed evaluation and discussion are presented in the subsequent section.

The MOWDO algorithm minimizes the pollution emissions and operational costs. All the generation units participated in supplying energy to the load. The grid is shown to provide excess energy during the daytime when demand is high. Still, as it is subjected to more carbon constraints, RESs, such as wind and solar systems, actively produce energy that reduces pollution emissions and increases power production. Furthermore, wind and solar systems have low operational costs, thus contributing to achieving both objectives simultaneously. From simulation results, it can be deduced that the optimal values of the developed MOWDO for pollution emission and operational cost are $316,400\mathrm{Kg}/\mathrm{kWh}$ and $339.4$ USD, respectively, as shown in Figure 10. In contrast, the existing MOPSO algorithm optimal values of pollution emission and operational cost are $336,400\mathrm{kg}/\mathrm{KWh}$ and $382.4$ USD, respectively, as illustrated in Figure 11. The proposed MOWDO and existing MOPSO numerical findings are presented in

Table 4. Numerical outcomes of the operational and emission optimization obtained through the developed MOWDO and existing MOPSO algorithm.

Techniques	Operational Cost	Pollution Emissions
MOWDO	339.4	316.5
MOPSO	382.4	336.5

. The above findings show that the proposed MOWDO algorithm best suits this problem and reduces the cost and emissions by 11.91% and 6.12%, respectively, compared to the existing MOPSO algorithm.

5. Conclusions

This study developed an optimization model considering both the generation side (DGs, diesel generators, batteries, and utility) and the demand side (residential, commercial, and industrial) for two objectives: pollution emission and operational cost optimization using multi-objective techniques. The consumers in the developed model are engaged in DR programs as coverage for the uncertainties associated with load demand, and PDF is used to handle the uncertainty associated with solar and wind generation. The developed optimization model using the MOWDO was implemented in MATLAB to solve the multi-objective problem via a fuzzy mechanism for both objectives. For validation, the developed model utilizing the MOWDO approach was compared to the MOPSO. The findings reveal that the MOWDO algorithm reduced pollution emissions and the operational cost by 6.12% and 11.91%, respectively, compared to the MOPSO algorithm. Hence, in resolving the multi-objective optimization problem concerning operational expense and pollution emissions, the MOWDO algorithm performs better than the MOPSO algorithm.