1 Introduction

Optimising the distribution of production among generation units is a key strategy in economic load dispatch (ELD). The objective is to minimise the overall generation costs while ensuring adherence to all fair and equality constraints. By properly allocating a portion of the overall demand, it is possible to reduce fuel expenditure. The load requirement is distributed among multiple generators, which affects estimation, unit commitment, billing, and various other processes. The entire volume of power generated must fully meet the current demand. In order to tackle this issue, the ELD may be categorised into two unique groups according on the load’s condition. Resource allocation to meet a predetermined demand is accomplished through two techniques in economic load dispatch: dynamic economic load dispatch (DELD) and static economic load dispatch (SELD). It’s important to address the same problem when dealing with a variable load requirement. Optimal results can be obtained by utilising dynamic economic load dispatch (DELD), a method that forecasts future demand and allocates it among different generators. Current reports suggest that researchers are actively working on integrating renewable energy sources into ELD systems, with the goal of addressing the problem of diminishing fossil fuel supplies. Traditional approaches such as the Lagrange multiplier fail to consider real-world constraints, resulting in a fitness function that is both non-linear and non-convex. To address this challenge, the utilisation of metaheuristic swarm optimisation techniques and artificial intelligence are crucial.

The study conducted by (Xu et al., 2014) investigated the use of genetic algorithm and dynamic programming in the Economic Load Dispatch (ELD) problem for 26 hydro power facilities in the three DP. In addressing the challenges of ELD and CEED within an isalanded microgrid system, Trivedi et al. (Trivedi et al., 2018) have exhibited the superior effectiveness of the Interior Search Algorithm (ISA) when compared to alternative methods such as the Reduced Gradient Method (RGM), Ant Colony Optimisation (ACO), and Cuckoo Search Algorithm (CSA). The Improved Sine Cosine Algorithm (ISCA) has been found to be more effective than the Cuckoo Search Algorithm (CSA), the Reduced Gradient Methods (RGM), and the Ant Colony Optimisation (ACO) when solving the identical objective function (Trivedi et al., 2015). The swarm optimisation algorithm, as described in the study by Azizipanah-Abarghooee (Azizipanah-Abarghooee, 2013), utilises a conflict-based hybridised Bacterial Foraging (BF) operation for different test systems with 5/10/30/100 units. Additionally, it incorporates a unique transformation of operators that considers the Prohibited Operating Zone (POZ) and Valve Point Effect (VPE). The investigation of communication issues utilizing 10 units of valve points has a more significant impact on determining the efficiency of the method. Maulik and Das (Maulik & Das, 2017) devised an innovative solution to tackle the problem of Economic Dispatch (ED) in microgrids. Their methodology combines lambda iteration, PSO, DSM optimisation methods, and lambda logic. The investigation showed that the lambda logic algorithm outperformed the other methods from every angle. According to Lahon and Gupta (Lahon & Chandra Prakash Gupta, 2018), utilities and microgrids may work well together in energy management partnerships. Reliable microgrid capability is available in both isolated and grid-connected configurations. According to Shrivastava and Nandrajog (Shrivastava & Saksham Nandrajog, 2017), classic optimization approaches such as Sequential Quadratic Programming (SQP), Interior Search Point Algorithms (ISPA), and other methods may be employed to handle Economic Dispatch (ED), which significantly reduces the cost of generation. Singh & Dhillon (Singh & Dhillon, 2019) proposed an ameliorated Grey Wolf Optimisation (GWO) method to handle economic power load dispatch problems that balance the exploration and exploitation. Roy (Roy, 2018) proposed an additional constraint to express the most significant possibility of coexisting variable demand and variable generation in static load dispatch. The additional constraint permits direct modeling of cross-correlations between changing power levels, making the dispatch suitable for high renewable power production situations. The authors of (Ma et al., 2017) presented a plug-in electric vehicle incorporated with ELD paradigm aimed at reducing generation costs and environmental pollution. This paradigm addresses the most challenging aspect of the system, as well as concerns related to cost in load dispatch. In order to satisfy the energy demand, the output power of all operational units may be balanced with the use of ELD assistance (Dai et al., 2021; Toopshekan et al., 2020). In order to assess the benefits of Demand Side Management (DSM) on the power generation side, Lokeshgupta and Sivasubramani (Lokeshgupta & Sivasubramani, 2018) has suggested a model that integrates Multi-Objective Dynamic Economic and Emission Dispatch (MODEED) with DSM. The suggested approach was validated using a test system consisting of six heat-producing units. Alham et al. (Alham & Elshahed, 2016) introduced a new DEED system that addresses the challenges of high wind penetration, intermittency, and uncertainty by including energy storage and using the direct search approach. Modified Differential Evolution (MDE) (Yuan et al., 2008), Improved Particle Swarm Optimisation (IPSO) (Yuan et al., 2009), Hybrid Evolutionary Programming-Sequential Quadratic Programming (EP-SQP) (Attaviriyanupap et al., 2002), and Hybrid Particle Swarm Optimization-Sequential Quadratic Programming (PSO-SQP) (Victoire et al., 2005), were all used to solve the economic dispatch problem for a system with ten units. This study showcases the importance of improving process conditions via the use of the experimental Taguchi technique and analysis of variance (ANOVA) (Abdelzaher, 2022; Elkhouly et al., 2021; Abdelzaher et al., 2022). By employing these statistical techniques, researchers can identify the most significant factors affecting the process and determine the optimal levels of these factors. The results of this study can help improve the efficiency and effectiveness of the process, leading to cost savings and improved product quality. Cost reduction is the primary goal of total power while considering any limit (Wu et al., 2021). Sun and Wang (Sun & Wang, 2017) presented a PSO with a velocity update based on a random dimension. After applying the proposed PSO to the power system’s dynamic economic dispatch issues, the algorithm’s performance is evaluated using a simulation of the benchmark functions upon which the algorithm is built. Daryani and Zare (Daryani, 2018) have ingeniously developed the heuristic Modified Group Search Optimisation (MGSO) algorithm, which exhibits remarkable capabilities in tackling the multi-objective combined economic power and emission dispatch problem. To understand the difficulties related to DELD, Daniel et al. (Daniel et al., 2018) proposed an ANN-based Levenbergh Marquardt Back-Propagation algorithm (LMBP). Considering the constraints of Ramp Rate Limit (RRL), the ELD problem was solved for nine generating units. In their study, Elsayed et al. (Elsayed et al., 2016) introduced an superior version of the social spider algorithm. The performance of the algorithm was evaluated on a test system consisting of the 6, 40, 80, and 140 test units. Liu and Nair (Liu et al., 2015) provided a new layout for a two-stage DELD model with stochastic nature using a disintegration strategy that was evaluated in RTS-24 (IEEE 24 Reliable Test System) and PJM-5(IEEE 5 Test System with generation parameters). To determine optimal scheduling for six different units with various transmission losses, a unique lambda iteration strategy has been employed by Chauhan et al. (Chauhan et al., 2017). A approach developed by Kumar and Dhillon (Kumar, 2018) employs a dynamically modified sequence of the Simplex Search Method (SSM) and the Artificial Algae Algorithm (AAA). When dealing with optimisation problems, AAA is often utilised as a global optimizer technique, while SSM employs a local search mechanism. In this article, the researcher implemented the Valve Point Effect (VPE) on a test system with 13-40-80 units. Jadoun et al. (Jadoun et al., 2018) introduced an innovative economic optimisation approach utilising the Improved Fireworks Algorithm with Non-Uniform Operator (IFA-NUO) cutting-edge algorithm. The innovative approach of incorporating an adaptive dimension strategy and implementing a limiting mapping operator and a non-uniform operator has been introduced. The researchers conducted a comprehensive analysis of the effectiveness of the proposed IFA-NUO approach within the framework of traditional dynamic economic load dispatch systems. Additionally, they successfully applied this technique to address the challenges posed by conventional DELD systems integrated with wind-solar systems. Ganjefar et al. (Ganjefar & Tofighi, 2011) have put forth a remarkable proposition in energy optimisation. They have introduced a cutting-edge genetic algorithm (GA) approach, complemented by a dynamic penalty function, to effectively address the intricate challenges posed by the dynamic economic dispatch problem. The same test system was then re-evaluated by incorporating a wind power plant by authors in (Yong & Tao, 2007). Table 1 provides a brief list of information about the recently published state-of-the-art CEED problems on dynamic systems.

Table 1 Exhaustive literature review based on state-of-the-art CEED problems
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1.1 Bridging the research gap with novel contribution

The innovative research being conducted on CEED problems in dynamic systems that account for numerous test systems and entities is demonstrated by the aforementioned review of the literature. Another factor that was noticed was that different types of CEED were utilised for different types of dynamic test systems in different publications. Most of the published CEED implementations use a Pareto front-based multi-objective approach. The literature review reveals a paucity of comparative analysis between multiple CEED methods and a lack of inquiry into why the multi-objective variety was chosen. Demand side management (DSM) is one economic strategy that can significantly reduce the generating cost of a distribution system by classifying and rescheduling elastic loads to off-peak times. By implementing DSM interventions at a distribution system, this study aims to close the research gap mentioned in the preceding section. Different forms of CEED have also been assessed on a 10-unit distributed system while taking valve point impact and DSM into account to integrate a more complex constraint situation. The primary goal of the article is to produce electricity in a method that reduces generation costs while maintaining a clean environment, i.e., releasing the least amount of hazardous emissions from the generators’ burning of fossil fuels. The tremendous exploration potential that JAYA offers, along with its ability to switch between the best and worst options, results in an adequate trade-off between the intensification and diversification processes. While CSA is distinguished by its high potential for exploitation, which guarantees that massive populations can be managed with ease and leads to quick convergence. In this article, a hybrid of these two algorithms called CSAJAYA was presented. This algorithm would guarantee adaptation of the greatest qualities of all three, resulting in optimum solutions.

1.2 Orientation of the manuscript

The remainder of the article is shaped in the methods mentioned below: Sect. 2 discusses the formulation of the goal function and discusses the specifics of CSAJAYA; Different assessment techniques are investigated, and the results are provided and discussed in Sect. 3. Section 4 presents the information’s outcome and conclusion.

2 Materials and methods

To minimise costs while satisfying all equity and efficiency requirements is the goal of economic load dispatch.

2.1 For DG units cost based fitness function

As stated in (Misra, Srikant et al., 2023; Dey et al., 2021), the cost function of fossil-fuelled generators is typically represented using a quadratic equation. The detailed equation for the cost function is shown below:

$${C_{DG}}=\sum\limits_{{t=1}}^{{24}} {\left[ {\sum\limits_{{i=1}}^{n} {({a_i}{P_{i,t}}^{2}+{b_i}{P_{i,t}}+{c_i})} +{C_{grid,t}}*{P_{grid,t}}} \right]}$$
(1)

Where a, b, c represents the cost coefficients for ith generator, Cgrid represents the TOU based pricing of the electricity. Pi and Pgrid denotes the power output of ith generator and grid respectively.

Equation (2) of the economic dispatch problem considers the valve-point effect, which is an important constraint for fossil-fueled generators as it involves the adjustment of valves to regulate the flow of steam and power (Dey et al., 2021). Figure 1 demonstrates that the valve-point effect introduces a sinusoidal characteristic, resulting in a cost function that has several modes.

Fig. 1
figure 1

Fuel curve of generating unit showing VPE

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$${C_{DG}}=\sum\limits_{{t=1}}^{{24}} {\sum\limits_{{i=1}}^{n} {({a_i}{P_{i,t}}^{2}+{b_i}{P_{i,t}}+{c_i}} +\left| {{d_i} \times \sin ({e_i}({P_{i,\hbox{min} }} - {P_{i,t}})} \right|} )$$
(2)

Pi is the ih unit’s output power in electrical. The valve point effect coefficients are di and ei, whereas the cost coefficients for the ith unit are ai, bi, and ci. As a direct result of this, the total cost is equal to CDG, provided that n is the number of DG units utilised.

2.2 DG unit emission dispatch

If we have the emission coefficients, we can plug them into (3 A) and (3B) to calculate the ED objective function. The ED equation includes an exponential component when VPE is taken into account, as illustrated in Eq. (3B), which creates the function multimodal. Where \({\alpha }_{i},{\beta }_{i},{\gamma }_{i},{\lambda }_{i}and{\epsilon }_{i}\) are the coefficients of emission of ith DG units whereas F2 is the total emission (Alham & Elshahed, 2016; Yuan et al., 2008, 2009).

$${F_2}=\sum\limits_{{t=1}}^{{24}} {\sum\limits_{{i=1}}^{n} {({\alpha _i}{P_{i,t}}^{2}+{\beta _i}{P_{i,t}}+{\gamma _i}} } )$$
(3A)
$${F_2}=\sum\limits_{{t=1}}^{{24}} {\sum\limits_{{i=1}}^{n} {({\alpha _i}{P_{i,t}}^{2}+{\beta _i}{P_{i,t}}+{\gamma _i}} } +{\lambda _i} \times \exp ({\varepsilon _i}{P_{i,t}}))$$
(3B)

2.3 Ppf based combined economic emission dispatch

In contrast to economic load dispatch’s focus on minimizing fuel costs, emission dispatch’s primary objective is to lower emissions of dangerous pollutants from conventional fossil fuel units. o mitigate fuel expenses and minimise environmental damage, it is essential to develop a novel approach. The “Price Penalty component” is used to generate a combined economic-emission dispatch (CEED) by integrating Eqs. (1), (2), (3A), and (3B). A single objective function (CEED) is obtained using this parameter (Alham & Elshahed, 2016; Yuan et al., 2008, 2009).

$$CEED=\sum\limits_{t}^{{24}} {\sum\limits_{{i=1}}^{n} {[({a_i}{P_{i,t}}^{2}+{b_i}{P_{i,t}}+{c_i}} } )\,+pp{f_i}*({\alpha _i}{P_{i,t}}^{2}+{\beta _i}{P_{i,t}}+{\gamma _i})]$$
(4A)
$$\begin{array}{l}CEED=\\\sum\limits_{t}^{{24}} {\sum\limits_{{i=1}}^{n} {[({a_i}{P_{i,t}}^{2}+{b_i}{P_{i,t}}+{c_i}} } +\left| {{d_i} \times \sin ({e_i}({P_{i,\hbox{min} }} - {P_{i,t}}))} \right|)\,+pp{f_i}*({\alpha _i}{P_{i,t}}^{2}+{\beta _i}{P_{i,t}}+{\gamma _i}+{\lambda _i} \times \exp ({\varepsilon _i}{P_{i,t}}))]\end{array}$$
(4B)

Mathematical expressions for the various price penalty factors are provided in (Yuan et al., 2008, Yuan et al., 2009).

2.4 FP based combined economic emission dispatch

This method attempts to reconcile the competing character of the two objective functions by expressing them as a ratio to each other, even if they share the same decision and control variables. Let’s say that Eq. (1) represents the ED equation, denoted as F1, and Eq. (2) represents the emission function, denoted as F2. Using the FP strategy, one can find an acceptable balance by reducing the F2:F1 ratio, as shown in Eq. (5) (Yuan et al., 2009).

$$FP=\frac{{\sum\limits_{{t=1}}^{{24}} {\sum\limits_{{i=1}}^{n} {({\alpha _i}{P_i}^{2}+{\beta _i}{P_i}+{\gamma _i}} )} }}{{\sum\limits_{{t=1}}^{{24}} {\sum\limits_{{i=1}}^{n} {({a_i}{P_i}^{2}+{b_i}{P_i}+{c_i}} )} }}$$
(5A)
$$FP=\frac{{\sum\limits_{{t=1}}^{{24}} {\sum\limits_{{i=1}}^{n} {({\alpha _i}{P_{i,t}}^{2}+{\beta _i}{P_{i,t}}+{\gamma _i}} } +{\lambda _i} \times \exp ({\varepsilon _i}{P_{i,t}}))}}{{\sum\limits_{t}^{{24}} {\sum\limits_{{i=1}}^{n} {({a_i}{P_{i,t}}^{2}+{b_i}{P_{i,t}}+{c_i}} } +\left| {{d_i} \times \sin ({e_i}({P_{i,\hbox{min} }} - {P_{i,t}}))} \right|)}}$$
(5B)

2.5 Mandatory constraints for the control variables

The problems in Eqs. (6) and (7), which are respectively issues without RES and problems with RES, have equality constraints. Equation (8) represents the inequality constraints that ensure the DERs remain within their designated boundaries.

$$\sum\limits_{{i=1}}^{n} {{P_{i,t}}} ={D_t}$$
(6)
$$\sum\limits_{{i=1}}^{n} {{P_{i,t}}+{P_{RES,t}}} ={D_t}$$
(7)
$${P_{i,\,min}} \leqslant {P_{i\,\,}} \leqslant {P_{i,\,max}}$$
(8)

Hence Dt represented as demand of tth hour. The power production of RES is denoted by.

PRES, t.

$$UP=\frac{{\sum\limits_{t} {{P_{i,t}}} }}{{24*{P_{i,\hbox{max} }}}}$$
(9)

UP stands for the utilisation percentage (Dey et al., 2021). Test structures with several DERs sometimes provide imprecise and confusing hourly results when presented in tables or figures, often denoted as UP.

2.6 Demand side management

The study of microgrid energy management, specifically in terms of maximising economic efficiency, has been a major and continuing field of research in recent years. It is expected to continue to be important in the future. In order to fully optimise the economic performance of a microgrid system, it is imperative to consider the strategic implementation of Demand Side Management (DSM) strategies. By implementing Demand Side Management (DSM), significant cost reductions can be achieved for the various articles examined in the comprehensive literature review (Dey et al., 2022). DSM priorities are shifting elastic loads on the studied network to cheaper times of day for the grid. Despite the total load demand near the edge of the scheduling horizon being constant, there is a significant drop in peak demand. This reduction in peak demand leads to a considerable increase in the load factor. Valley filling, strategic conversion, strategic expansion, peak clipping, load shifting, and variable load shape are some of the dynamic load management techniques utilized by Demand Side Management (DSM) to optimise energy consumption patterns. As explained in references (Dey et al., 2022; Basak et al., 2022), DSM makes use of a wide variety of load shaping tactics, such as valley filling, strategic conversion, strategic expansion, peak clipping, load shifting, and variable load shape.

Listed below are the specific measures that should be taken while putting DSM into practice:

Step 1: Provide the duration of the dynamic load data in terms of hours.

Step 2: Enter the TOU price of electricity on the market for Tth hours.

Step 3: Enter the DSM involvement % (For unspecified elastic loads).

Step 4: Figure out the inelastic and elastic loads based on the DSM involvement rate. When we say “x% DSM,” for example, it means that x% of the hourly load demand changes, but the other (100-x) % does not. It is more effective to carefully prepare for the constraints of the elastic load.

Step 5: Determine the lowest and maximum values, as well as the total, for the inelastic load. It is significant to acknowledge that the control variables that need to be optimised have a flexible load requirement.

Step 6: Applying the optimisation approach,

$$Minimize\,[{\text{cost}}_{{grid}}^{t}*(elastic\_load{\,^t}+inelastic\_load{\,^t})]$$
(10)

Where,

i)

$$0 \leqslant elastic\_load{\,^t} \leqslant elastic\_loa{d^{\hbox{max} }}$$
(11)

ii)

$${\text{D}\text{e}\text{m}\text{a}\text{n}\text{d}}^{{\prime }}\text{s} \text{t}\text{o}\text{t}\text{a}\text{l} \text{l}\text{o}\text{a}\text{d} ={\sum }_{t=1}^{T}\left(elasti{c}_{loa{d}^{t}}+inelasti{c}_{loa{d}^{t}}\right)$$
(12)

Step 7: The revised load demand model incorporates the demand-side management (DSM) approach, including the aggregate non-responsive load demand for each hour and the optimised values of responsive loads attained.

2.7 The proposed hybrid algorithm

This paper utilises a combination of the CSA (Askarzadeh, 2016; Dey et al., 2020) and JAYA (Rao & Jaya, 2016) algorithms as an intelligent optimisation tool for scheduling the DG outputs, with a focus on the previously mentioned fitness function. The Hybrid CSAJAYA is theoretically represented by replacing the random location of the crow with the standard JAYA equation. In the traditional CSA method, there was a potential danger of compromising the fitness function value and resulting in a suboptimal solution if the randomly generated value was lower than the awareness probability (AP). Nevertheless, by this adjustment, the amalgamation with JAYA ensures that the solution of the fitness function will consistently progress towards the optimal outcome without deviating inside the search area. Authors in (Karmakar & Bhattacharyya, 2020) implemented CSAJAYA algorithm for solving reactive power planning problem with FACTS devices after realizing the algorithm on benchmark function. The researchers in (Basak & Bhattacharyya, 2023) used a hybrid CSAJAYA approach to investigate the impact of various grid participation and pricing methods, as well as the effects of valve point loading and wind energy uncertainty, on the complexity and feasibility of the microgrid. The hybrid CSAJAYA method may be mathematically represented by Eq. (13). The memory update equation is identical to the CSA equation and is referenced as Eq. (14).

$${X^{u,iter+1}}=\left\{ \begin{gathered} {X^{u,iter}}+ran{d_1} \times fl. \times ({m^{v,iter}} - {X^{u,iter}})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,if\,ran{d_j} \geqslant AP\,\,\,\,\,\,\,\, \hfill \\ \,for\,k=1:n \hfill \\ {X^{u,iter,k}}+ran{d_1} \times m_{{best}}^{k} \times \left| {{X^{u,iter,k}}} \right| - ran{d_2} \times m_{{worst}}^{k} \times \left| {{X^{u,iter,k}}} \right|\,\,\,\,\,\,\,\,\,,otherwise \hfill \\ end \hfill \\ \end{gathered} \right.$$
(13)
$${m^{u,iter+1}}=\left\{ \begin{gathered} {X^{u,iter+1}},\;if\;f({X^{u,iter+1}})\;<\;f({m^{u,iter}}) \hfill \\ {m^{u,iter}},\,\,\,\,\,\,\,\,\,\,otherwise \hfill \\ \end{gathered} \right.$$
(14)

Furthermore, the value of AP is theoretically formulated to vary with each iteration, while always remaining within the range of 0 and 0.99, since it represents a probability. This eliminates the need for any tuning parameters and simplifies the execution of the algorithm.

2.8 CSA-JAYA algorithm for dynamic load dispatch

The population is determined by the variables in Eq. (15), and each parameter of the population remains inside the boundaries of the system as shown in Eq. (6) to (8). Given a population size N, a distribution set size D, and an optimal scheduling time T, (15).

$$X=\left[ \begin{gathered} X_{{1,DER1}}^{1},\;X_{{1,DER1}}^{2},{\cdots}\;X_{{1,DER1}}^{T},\;X_{{1,DER2}}^{1},\;X_{{1,DER2}}^{2},{\cdots}\;X_{{1,DER2}}^{T}{\cdots}X_{{1,DER\;D}}^{1},\;X_{{1,DER\;D}}^{2},{\cdots}\;X_{{1,DER\;D}}^{T} \hfill \\ X_{{2,DER1}}^{1},\;X_{{2,DER1}}^{2},{\cdots}\;X_{{2,DER1}}^{T},\;X_{{2,DER2}}^{1},\;X_{{2,DER2}}^{2},{\cdots}\;X_{{2,DER2}}^{T}{\cdots}X_{{2,DER\;D}}^{1},\;X_{{2,DER\;D}}^{2},{\cdots}\;X_{{2,DER\;D}}^{T} \hfill \\ X_{{3,DER1}}^{1},\;X_{{3,DER1}}^{2},{\cdots}\;X_{{3,DER1}}^{T},\;X_{{3,DER2}}^{1},\;X_{{3,DER2}}^{2},{\cdots}\;X_{{3,DER2}}^{T}{\cdots}X_{{3,DER\;D}}^{1},\;X_{{3,DER\;D}}^{2},{\cdots}\;X_{{3,DER\;D}}^{T} \hfill \\ {\cdots}{\cdots}{\cdots}{\cdots}. \hfill \\ {\cdots}{\cdots}{\cdots}{\cdots}. \hfill \\ X_{{N,DER1}}^{1},\;X_{{N,DER1}}^{2},{\cdots}\;X_{{N,DER1}}^{T},\;X_{{N,DER2}}^{1},\;X_{{N,DER2}}^{2},{\cdots}\;X_{{N,DER2}}^{T}{\cdots}X_{{N,DER\;D}}^{1},\;X_{{N,DER\;D}}^{2},{\cdots}\;X_{{N,DER\;D}}^{T} \hfill \\ \end{gathered} \right]$$
(15)

The crows represent the components of the population matrix that operate as the control variables, and their position is directly correlated with the value of the fitness function. The term “murder” refers to the whole population matrix characterized by Eq. (15). The crow’s location in the search space that produces the smallest value for the fitness function is considered the best solution within the framework of a constrained minimization method. This position is often known as the “preeminent position within a group of crows”. Figure 2 visually represents the method mentioned before.

Fig. 2
figure 2

Flowchart for CSA-JAYA algorithm

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3 Results and discussions

3.1 Preliminary description of the subject and test system and software

The subject test system consists of 10 fossil fueled generators supported by a wind farm. Tables 2, 3 and 4 display the magnitudes associated with DG parameters. Hourly load demand and wind power output are shown in Figs. 3 and 4, respectively. The optimisation process was performed using a desktop computer equipped with 8GB of RAM and an Intel-i5 CPU operating at a frequency of 2.7 GHz. A population size of 80 was taken into account while calculating all of the fitness functions in order to conduct an objective investigation. A limit of 500 iterations was established.

Fig. 3
figure 3

Load demand curve

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Fig. 4
figure 4

Wind power output

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3.2 Test system

The paper’s entire process is divided into three stages, which will be explained in detail below:

3.2.1 Stage 1

At this stage, demand side management is used to modify the predicted day-ahead load demand of the dynamic subject test system. Assuming that 40% of the total hourly load demand can be moved to cheaper times of day, optimal scheduling can be used to move these loads. DSM takes into account hourly electricity rates collected from (Abdollahi et al., 2011). Figure 5 displays both the original predicted load demand as well as the reorganized load demand structure that resulted from the adoption of 40% DSM. Table 5 shows some of the most important perks of adopting the DSM, including:

Fig. 5
figure 5

Load demand with and without DSM

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  1. a.

    Before and after DSM was put in place, the total load demand at the end of the day (40,108 kW) stayed the same.

  2. b.

    Before and after DSM was put in place, the average load demand at the end of the day was 1671 kW.

  3. c.

    As a result of putting in place DSM, peak demand dropped by 12%, from 2220 kW to 1953.5 kW.

  4. d.

    The system’s load factor went from 0.7528 to 0.85, which is better.

3.2.2 Stage 2

At this point, the subject test system’s generation cost was minimised using the proposed CSAJAYA algorithm with and without the wind farm’s power being factored in. The minimized generation cost obtained by the proposed algorithm was $1,012,182 without wind and $977,760 with wind. The same can be seen in Table 6. Furthermore, when the same was evaluated for restructured load demand after the implementation of DSM, the generation cost was reduced to $1,007,441 and $972,874 without and with wind correspondingly. Table 6 shows that the suggested CSAJAYA algorithm achieved the lowest generation cost for the examined test system, besting a broad variety of algorithms. The distribution of load among 10 generators in the presence of wind is illustrated in Figs. 6 and 7. In one scenario, the generation cost was minimized with DSM, while in the other scenario, it was minimized without DSM.

Fig. 6
figure 6

Hourly output ECD with wind-without DSM

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Fig. 7
figure 7

Hourly output ECD with wind-with DSM

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3.2.3 Stage 3

Combined Economic Emission Dispatch (CEED): Two types of CEED was analyzed for the subject test system as discussed in Sect. 2 of the manuscript. For PPF based CEED, initially min-max penalty factor was evaluated for all the 10 generators as per the formula mentioned earlier and in (Dey et al., 2020) and are displayed in Table 7. Thereafter, the fitness function corresponding to CEEDPPF was evaluated and the total results obtained are listed in Table 8. The penalized cost for CEEDPPF with wind as minimized using CSAJAYA algorithm was $1,231,435 without DSM. This was further reduced to $1,216,885 when the same was assessed for DSM implemented re-organized load demand. The costs obtained using proposed algorithm is lesser than five different algorithms as reported in literature as seen from Table 8.

Thereafter, FP based CEED was assessed using the proposed algorithm without and with think about DSM. The value of the fitness function for CEEDFP, as expressed by Eq. 5B was 0.23709 without DSM and 0.2065 with DSM respectively. Both costs and emission were than evaluated for both the methods of CEED with and without DSM. Table 9 demonstrates that the inclusion of DSM based load demand resulted in lower generating cost and emissions. Furthermore, it can be also noticed from the table that FP based CEED yielded a better trade off explanation between generation cost and emission when matched to PPF based CEED. Figures 8 and 9 showing the hourly based output of the generators when CEEDPPF and CEEDFP was evaluated using proposed algorithm considering DSM.

Fig. 8
figure 8

Hourly output CEEDPPF with DSM

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Fig. 9
figure 9

Hourly output CEEDFP with DSM

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Figure 10 shows the convergence curve characteristics of the generation cost was estimated with and without DSM using the suggested hybrid algorithm. The early convergence property of the algorithm can be realized from this figure. The statistical data from Table 10 indicates that the recommended approach outperforms several alternative algorithms discussed in terms of both minimal standard deviation and elapsed time.

Fig. 10
figure 10

Convergence curve characteristics

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4 Conclusions

4.1 Implications to theory and key lessons learnt

The 10-unit non-linear dynamic test system that was setup with RES was used to compare and contrast two distinct techniques of measuring CEED in this article. DSM usage not only lowers peak demand and improves the system’s load factor, but it also creates a load demand model that is very cost-effective. This was witnessed by the values of all the fitness functions evaluated in this work. CEEDFP yielded a well-furnished solution between emission and generation cost as compared to CEEDPPF. CSAJAYA consistently produced higher-quality results than a variety of other optimization techniques. This is a good reason to use the mixed optimization method in the future to solve large-scale, hard optimization problems.

4.2 Limitations of the research and scope to expand the horizon of the present research

The same work may be done while taking ramp rates, start-up/shut down costs and transmission losses into account, which were not considered in this study. This would increase the study’s complexity and usefulness. These test systems can also be used to address the logistical challenges of generating units that are habitually run at greater than 80–90% of their capacity, such as those posed by electric vehicles integrated into the microgrid system and the placement of charging stations.