1 Introduction

1.1 Motivation

Increasing concern about global warming and depletion of fossil fuel sources have emphasized the importance of electricity generation from Renewable Energy Sources (RESs) [1,2,3,4,5] and the integration of Energy storage system with their high efficiency and stable operation [6, 7]. So the integration of these RESs as sub-systems production or Distributed Generation (DG), which is known as Microgrids (MGs), into the main grid is essential for solving energy-related issues [8]. The advantages of incorporating RESs into the MG include lower operating costs, higher profitability for investors, reduced level of greenhouse gas emissions, and cheaper power obtained from the grid [9].

The MG has three distinct advantages: (1) Technical advantages, such as enhancing energy efficiency and increasing power system resilience; (2) Economic benefits, such as reducing fuel and interruption costs; and (3) Environmental benefits, such as reduced greenhouse gas emissions [10].

EM in MG is considered as operation and planning stage, which is considered a cost-driven scheduling for MG sources, to supply load demand with maintaining some constraints [11]. Moreover, with the advancements in smart grid technology, there has been a rise in the load demand [12], thereby necessitating the implementation of an energy management system to ensure a balance between load and energy supply [13]. As a result, there has been a significant research focus on MG Energy Management to achieve optimal operation of MGs [14,15,16,17,18,19].

Although peak hours are just a few hours per day, significant investment in generation, transmission, and distribution infrastructure is necessary to supply peak demand. As a consequence, the cost of power supply increases. So, Demand Side Management programs (DSMP) are integrated into EM systems in MGs, which can enhance the load curve by allowing the interpolation of the customers [20].

These DSMP schemes can be categorized into two main groups energy efficiency and Demand Response Programs, with further sub-categories, incentive-based DR (IB-DR) and price-based DR(PB-DR), as shown in Fig. 1 [21].

Fig. 1
figure 1

Categorization of demand side management techniques

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1.2 Related work

In [22], using Particle Swarm Optimization (PSO), a co-optimization technique for MG planning was developed to identify the appropriate sizing of DG sources while optimizing yearly fuel costs. To maximize solar-based MGs' techno-economic and environmental advantages [18] presents a MG model that aims for maximum reliability. In Ref. [23], the authors suggested an Ant Colony Optimization (ACO) technique for solving the EM and determining optimal scheduling for energy sources in the MG in order to lower the overall generation cost. Optimal scheduling for generation sources to get economic benefits of MG comprised of RESs and Energy storage system (ESS) is proposed using PSO in [24] and improved PSO in [25]. In [26] the optimal operation of MG based on Water Cycle Algorithm (wCA) is solved to minimizing the operating cost. In [27] a Mixed Integer Distributed Ant Colony Optimization is developed to solve the MG economic dispatch problem for cost minimization. According to the aforementioned works, the main research considered passive customers' participation in solving EM in MG. Implementing solutions to alleviate the network's demand side may help further optimize the Microgrid's EM problem.

In literature, different types of DRPs have been discussed in order to optimize the MG operation. [28] Used a hybrid augmented weighted ε-constraint technique and lexicographic optimization to solve optimal power flow in MG considering DR in a combined heat and power (CHP) system. Annual operating cost is minimized in [29] while maximizing customer satisfaction. In [30], different types of DRP have been introduced and compared for Multi-Microgrid using linear programming with Mixed integer mathematical models. In [31], optimal scheduling for the MG with RES considering its stochastic nature and conventional generators has been presented. In addition, an IB-DR program has been used to minimize energy cost and ensure customers' benefits using the technique of weighted sum and the fuzzy satisfying method. An IB-DR for deterministic EM was proposed in [32] using Honey Badger Optimizer, and Probabilistic EM using Artificial Hummingbird Algorithm in [33] A PB-DR is formulated in [34] to minimize the cost of power losses in MG and in [35] using PSO to maximize the customers’ profit. In [36] a multi-objective EM problem is solved based on Genetic algorithm (GA) using PB-DR to minimize the net present cost. In [19] a formulation for an IB-DR to minimize the generation cost and maximize the MG operator is proposed. While Ref. [37] solve the economic dispatch problem based on IB-DR using Genetic algorithm Ref. [38] a hybrid technique for EM is implemented based on Pelican Optimization Algorithm to solve the optimal operation and to reduce the demand during the peak periods. In [39] an IB-DR has been used for bi-level programing; in the upper-level based on unit commitment the marginal prices is achieved, and the cost is minimized in the lower-level. A multi-objective DR problem using α-constrained dominated differential evolution algorithm for reducing the customers’ bill and enhance the energy efficiency is proposed with integration of DR [40].

According to the above-mentioned researches, many optimization approaches have effectively handled the engineering problems, particularly those involving EM. The findings of these studies demonstrated the importance and potential for developing new and improved optimization approaches for solving specific EM problems; also According to the No-Free-Lunch (NFL) theorem [41] there are either no metaheuristic optimization algorithms capable of solving all optimization problems. These two reasons motivate us to propose a new optimization technique, called Leader-based mutation-selection INFO (LINFO) based on the weIghted meaN oF vectOrs (INFO) algorithm to prevent the possibility to be trapped into local minima. We chose the algorithms presented here due to their outstanding performance in solving a variety of mathematics and engineering design challenges.

1.3 Contribution

The first objective of this paper is to propose an improved version of INFO algorithm to overcome its weakness; the second objective is to optimize MG's operation considering incentive DR based on the proposed algorithm and compare its performance with other optimization metaheuristic and newly developed algorithms.

The main contributions of this paper to the aforementioned investigation may be stated as follows:

  1. 1.

    Proposing an improved algorithm called LINFO to avoid the drawbacks of the original INFO algorithm of being trapped in local optima. And Validating the performance of the proposed algorithm by comparing it with newly developed algorithms to validate its performance, based on fitness value using different test benchmark functions based on different a statistical terms.

  2. 2.

    Solving MG energy management problem considering demand response program based on the proposed LINFO algorithm. The performance of the proposed LINFO algorithm is compared with well-known and newly developed algorithms in solving the EM problem.

1.4 Paper organization

The rest of this paper is structured as follows: Sect. 2 presents modeling of the Grid-connected Microgrid with modeling of DRP. Section 3 discusses the EM optimization problem modeling; Sect. 4 focuses on the proposed LINFO algorithms. The obtained simulation results are presented in Sect. 5. Finally, the paper is concluded in Sect. 6.

2 Grid-connected MG scheme

2.1 Modeling

  • MG model

The structure of proposed MG is shown in Fig. 2. MG's resources are a PV system, WT system, non-renewable sources such as DE and customers with DRP. Integrating DG (including renewable sources) requires power conversion units; renewable sources need a power electronics converter to track the maximum power and synchronization with the grid [42]. In this study, it is assumed that power is transacted between the MG and the main grid. Cost of Power transacted (\(P_{{g_{t} }}\)) at any time interval t is \(C_{g} \left( {P_{{g_{t} }} } \right)\) and can be obtained as [43]:

$$C_{g} \left( {P_{{g_{t} }} } \right) = \left\{ {\begin{array}{*{20}l} {\gamma_{t} \times \left| {P_{{g_{t} }} } \right|} \hfill & {P_{{g_{t} }} > 0} \hfill \\ 0 \hfill & {P_{{g_{t} }} = 0} \hfill \\ { - \gamma_{t} \times \left| {P_{{g_{t} }} } \right| } \hfill & {P_{{g_{t} }} < 0} \hfill \\ \end{array} } \right\}$$
(1)

  • Solar power modeling

Fig. 2
figure 2

Grid-connected MG system

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For a given area, the hourly output power from the solar PV generator is given as [44]:

$$P_{st} = \eta_{{{\text{PV}}}} A_{c} I_{{{\text{Pv}}_{t} }}$$
(2)

where, \(\eta_{{{\text{pv}}}}\) is solar PV system’s efficiency, which is a function of the incident solar irradiation \(\left( {I_{{{\text{Pv}}_{t} }} } \right){ }\)(kW h/m2), and the ambient temperature and \(A_{c}\) is the PV array area [44].

  • WT modeling

The hourly power produced by a WT (\(P_{{{\text{wt}}}}\)) is entirely dependent on hourly wind speed at a specific height \((v_{{{\text{hub}}_{t} }} )\), air density, swept area and converter efficiency. Using power-law equation, wind speed at the required hub height may be converted from the anemometer height as following [45]:

$$v_{{{\text{hub}}_{t} }} = v_{{{\text{ref}}_{t} }} \left( {\frac{{h_{{{\text{hub}}}} }}{{h_{{{\text{ref}}}} }}} \right)^{\beta }$$
(3)

where \(h_{{{\text{hub}}}}\) is hub height; \(v_{{{\text{ref}}_{t} }}\) is the hourly wind speed at the reference hub height \(h_{{{\text{ref}}}}\). The power law exponent \(\beta\) is usually is in the range between \(\frac{1}{4}{ }\) and \({ }\frac{1}{7}\).

The hourly wind power is calculated as [45]:

$$P_{{{\text{wt}}}} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {v \le v_{{{\text{ci}}}} \;{\text{and}}\;v \ge v_{{{\text{co}}}} } \hfill \\ {\frac{{v_{{{\text{hub}}_{t} }}^{2} - v_{{{\text{ci}}}}^{2} }}{{v_{{{\text{nom}}}}^{2} - v_{{{\text{ci}}}}^{2} }}.P_{{{\text{nom}}}} } \hfill & {v_{{{\text{ci}}}} \le v_{{{\text{hub}}_{t} }} \le v_{{{\text{nom}}}} } \hfill \\ {{\text{P}}_{{{\text{nom}}}} } \hfill & {v_{{{\text{nom}}}} < v_{{{\text{hub}}_{t} }} \le v_{{{\text{co}}}} } \hfill \\ \end{array} } \right.$$
(4)

where \(P_{{{\text{nom}}}}\) is the rated power of WT, \(v_{{{\text{nom}}}}\) is the rated wind speed,\(v_{{{\text{co}}}}\) the cut-out wind speed, and \(v_{{{\text{ci}}}}\) the cut-in wind speed, and, with \(P_{w}\) and \(v\) denoting output power and wind speed.

  • Modeling of DR

The formulation for customer's cost function \(\left( {C\left( {\theta ,x} \right)} \right)\) can be obtained mathematically as [46]:

$$C\left( {\theta ,x} \right) = k_{1} x^{2} + k_{2} x\left( {1 - \theta } \right)$$
(5)

where \(k_{1}\), and \(k_{2}\) are cost coefficients, and \(\theta\) is the customer willing rate; it ranges between 0 and 1, with the most willing Costumer having a value of 1 and the least willing customer having a value of 0.

Customers will reduce their consumption in response to DRP only if their benefit \(B_{j} \ge 0{ }\) and so the benefit for any customer j can be expressed as [43]:

$$B_{j} \left( {\theta ,y,x} \right) = y_{j} - \left( {k_{1} x_{j}^{2} + k_{2} x_{j} - k_{2} x_{j} \theta } \right),\quad {\text{for}}\;j = 1,2, \ldots J$$
(6)

where \(y_{j}\) is the incentive that customers receive, \(x_{j}\) is the amount of consumption reduction in (KW or MW).

Moreover, MG benefit is calculated as [43]:

$$B_{0} = \mathop \sum \limits_{j = 1}^{J} \lambda_{j} x_{j} - y_{j} .$$
(7)

3 The operating cost function

As previously stated, the Grid-connected MG is composed of various generating sources, including conventional generators and RESs, as well as load with a DRP. The operational cost is composed of two components: the cost of DEs generation and the cost of power transactions. So, the EM system’s objective in MG is to optimally operate MG’s resources by minimizing generation costs and maximization MG benefit while meeting operational restrictions.

Thus, the formulation of the EM problem may be described mathematical as follows:

$$\min f_{1} \left( x \right) = \min \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{i = 1}^{I} C_{i} \left( {P_{{i_{t} }} } \right) + \mathop \sum \limits_{t = 1}^{T} C_{g} \left( {P_{{g_{t} }} } \right)$$
(8)

While \(C_{i} \left( {P_{{i_{t} }} } \right)\) is the \(i{\text{th}}\) conventional generator’s fuel cost at any time interval \(t\) and it is represented by a quadratic model as follows [44]:

$$C_{i} \left( {P_{{i_{t} }} } \right) = a_{i} p^{2}_{{i_{t} }} + b_{i} P_{{i_{t} }}$$
(9)

where ai and bi are fuel cost coefficients.

From (9), it is seen that the first term in this first objective function includes the total operating cost of the DE units. While the second term represents the cost of power transaction between the main grid and MG.

The second objective of EM system is maximizing the MG operator benefit, which is expressed as in (7); the objective function can be calculated as

$$\max f_{2} \left( x \right) = \max \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{i = 1}^{I} \lambda_{j} x_{j} - y_{j}$$
(10)

So that the mathematical model of the objective function for MG management is given as:

$$\min w_{1} \left[ {\left. {\mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{i = 1}^{I} C_{i} \left( {P_{{i_{t} }} } \right) + \mathop \sum \limits_{t = 1}^{T} C_{r} \left( {P_{{r_{t} }} } \right) } \right]} \right. + w_{2} \left[ {\left. {\mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{i = 1}^{I} y_{j} - \lambda_{j} x_{j} } \right]} \right.$$
(11)

where w is the weighting factor and the following equation should be satisfied:

$$w_{1} + w_{2} = 1.$$
(12)

3.1 Operating constraints

  • Power balance constraints [43]:

    $$\mathop \sum \limits_{i = 1}^{I} P_{{i_{t} }} + P_{wt} + P_{st} + P_{{r_{t} }} = D_{t} - \mathop \sum \limits_{j = 1}^{J} x_{j,t} .$$
    (13)

The total generated power from MG sources (DEs, WT, and PV), and power transacted should match the load demand.

Where \(D_{t}\) and \(x_{j,t}\) are the total demand and the \(j{\text{th}}\) customer power curtailed at time t, respectively.

  • Generation constraints [43]:

    $$P_{{i_{\min } }} \le P_{{i_{t} }} \le P_{{i_{\max } }}$$
    (14)
    $$- DR_{i} \le P_{{i_{t + 1} }} - P_{{i_{t} }} \le UR_{i}$$
    (15)
    $$0 \le P_{{s_{t} }} \le P_{{st_{\max } }}$$
    (16)
    $$0 \le P_{{w_{t} }} \le P_{{wt_{\max } }}$$
    (17)

where \(P_{{i_{\min } }}\) and \(P_{{i_{\max } }}\) are the minimum and maximum power generated from any generator \(i\), respectively; \(UR_{i}\) and \(DR_{i}\) are the maximum ramp up and ramp down rates for generator I;\(P_{{st_{\max } }}\) and \(P_{{wt_{\max } }}\) are the forecasted maximum power from PV and wind generators at any time \(t\).

Constraint (14) ensures that any DE generation is within the maximum and minimum limits, while constraint (15) states that the ramping up and down rate limits should not be violated. Equations (16) and (17) represent the constraints of the maximum and minimum generation limits of WT generator and solar PV generation, respectively.

  • Power transaction constraints:

The power transacted power between utility grid and MG should not exceed the maximum permissible limit \(P_{{g_{\max } }}\) as [43]:

$$- P_{{g_{\max } }} \le P_{{g_{t} }} \le P_{{g_{\max } }}$$
(18)

  • Demand response constraints [43]:

The demand management formulation in (6) is extended for the whole day; so customers' benefit can be expressed as in (19) to ensure the incentive is higher than the cost that the customer should pay.

$$\mathop \sum \limits_{t = 1}^{T} y_{j,t} - \left( {k_{1} x_{j,t}^{2} + k_{2} x_{j.t} - k_{2} x_{j,t} \theta } \right) \ge 0$$
(19)
$$\mathop \sum \limits_{t = 1}^{T} y_{j,t} - (k_{1} x_{j,t}^{2} + k_{2} x_{j,t} - k_{2} x_{j,t} \theta ) \ge \mathop \sum \limits_{t = 1}^{T} y_{j - 1,t} - (k_{1} x_{j - 1,t}^{2} + k_{2} x_{j - 1,t} - k_{2} x_{j - 1,t} \theta )\quad {\text{For}}\;j = 2,3, \ldots ,J$$
(20)
$$\mathop \sum \limits_{t = 1}^{J} x_{j,t} \le {\text{CM}}_{j}$$
(21)
$$\mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{j = 1}^{J} y_{j,t} \le {\text{UB}}$$
(22)

where \({\text{UB}}\) and \({\text{CM}}_{j}\) are daily MG budget limit and customer j daily power curtailment limit.

Constraint in (20) ensures that the bigger the customer's curtailment, the larger the incentive they will get. The total MG budget limit constraint is described in Eq. (21) to ensure the daily budget is lower than the limit. Equation (22) is to ensure the total curtailment of any customer j is within the permissible limits.

4 Solution method

In this section, the weIghted meaN oF vectOrs (INFO) algorithm is briefly presented then the process of the proposed leader INFO (LINFO) technique is described.

4.1 Original INFO algorithm

The INFO algorithm was introduced in 2022 [47]. For updating the vectors’ location, INFO algorithm use four main processes: initialization stage, updating rule, vector combining, and a local search.

  • Initialization stage

The INFO algorithm is composed of n vectors population in search space with diminution D. The initial population is generated at random as following:

$$X_{n} = X_{\min } + rand\left( {0,1} \right).\left( {X_{\max } - X_{\min } } \right)$$
(23)

where \(X_{n}\) denotes the nth vector, \(X_{\min } , X_{\max }\) are the bounds of the solution domain in each problem and \(rand\left( {0,1} \right)\) return a random number in an interval of [0, 1].

  • updating rule stage

This stage enhances the diversity of the population throughout the search process. This operation creates new vectors using the weighted mean of existing vector. The main formulation of the updating rule is presented as:

(24)

where \(z1_{l}^{g}\) and \(z2_{l}^{g}\) are the new vectors in the gth generation; and \(\sigma\) is the scaling rate of a vector, and it is defined as:

$$\begin{gathered} \sigma = 2\alpha \times rand - \alpha \hfill \\ \alpha = 2 \times \exp \left( { - 4.\frac{g}{Maxg}} \right) \hfill \\ \end{gathered}$$
(25)

  • vector combining stage

In order to enhance the diversity of the population in the INFO algorithm, using the \(z1_{l}^{g}\) and \(z2_{l}^{g}\) that calculated in the previous stage (new vectors) in a combination with vector \(x_{l}^{g}\) based on the following equations as:

figure a

Where \(\mu\) have a value of \(0.05 \times randn\).

  • local search stage

The search ability of this stage prevents the algorithm to drop into local optima. Considering the local operator using the global position (\(x_{\text{best}}^{g}\)), a new vector can be generated around this global position as follows:

figure b

In which

$$x_{{{\text{rnd}}}} = \phi \times x_{{{\text{avg}}}} + \left( {1 - \phi } \right) \times \left( {\phi \times x_{bt} + \left( {1 - \phi } \right) \times x_{bs} } \right)$$
(27.3)
$$x_{{{\text{avg}}}} = \frac{{\left( {x_{a} + x_{b} + x_{c} } \right)}}{3}$$
(27.4)

where \(\phi\) is a random number in between 0 and 1; and \(x_{rnd}^{{}}\) denotes a new solution that randomly combines the three solutions (\(x_{{{\text{avg}}}}\), \(x_{bt}\), and \(x_{bs}\)); Which increase proposed algorithm’s nature of randomness, and so better search ability in the search space. \(\upsilon_{1}\) and \(\upsilon_{2}\) are two random numbers given by the following equations:

$$\upsilon_{1} = \left\{ {\begin{array}{*{20}l} {2 \times rand} \hfill & {{\text{if}}\;p > 0.5} \hfill \\ 1 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$$
(27.5)
$$\upsilon_{2} = \left\{ {\begin{array}{*{20}l} {rand} \hfill & {{\text{if}}\;p < 0.5} \hfill \\ 1 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$$
(27.6)

where p is a random number in the range of (0, 1).

4.2 Proposed leader INFO algorithm

The development of the Leader-based mutation-selection [48] to solve the possibility of the optimal value may drop into a local minima. This modification depends on the best vector’s position \(x_{{{\text{best}}}}^{t}\), the following best vector’s position (second-best) \(x_{{{\text{best}} - 1}}^{t}\) and the third-best vector position vector \(x_{{{\text{best}} - 2}}^{t}\) based on the value of the objective function on the new vector’s location \(x_{i} \left( {{\text{new}}} \right)\) between the number of population. Then the new mutation vector’s position vector \(x_{i} \left( {{\text{mut}}} \right)\) is given by:

$$x_{i} \left( {{\text{mut}}} \right) = x_{i} \left( {{\text{new}}} \right) + 2 \times \left( {1 - \frac{t}{Max\_it}} \right) \times \left( {2 \times rand - 1} \right)\left( {2 \times x_{{{\text{best}}}}^{t} - \left( {x_{{{\text{best}} - 1}}^{t} + x_{{{\text{best}} - 2}}^{t} } \right)} \right) + \left( {2 \times rand - 1} \right)\left( {x_{{{\text{best}}}}^{t} - x_{i} \left( {{\text{new}}} \right)} \right)$$
(28)

Then, the next location is updated using the following equation:

$$x_{i} \left( {t + 1} \right) = \left\{ {\begin{array}{*{20}c} {x_{i} \left( {{\text{mut}}} \right) \, f\left( {x_{i} \left( {{\text{mut}}} \right)} \right) < f\left( {x_{i} \left( {{\text{new}}} \right)} \right)} \\ {x_{i} \left( {{\text{new}}} \right) \,f\left( {x_{i} \left( {{\text{mut}}} \right)} \right) \ge f\left( {x_{i} \left( {{\text{new}}} \right)} \right)} \\ \end{array} } \right.$$
(29)

Finally, the best solution is updated as follows:

$$x_{{{\text{best}}}} = \left\{ {\begin{array}{*{20}c} {x_{i} \left( {{\text{mut}}} \right) \quad f\left( {x_{i} \left( {{\text{mut}}} \right)} \right) < f\left( {x_{{{\text{best}}}} } \right)} \\ {x_{i} \left( {{\text{new}}} \right) \quad f\left( {x_{i} \left( {{\text{new}}} \right)} \right) < f\left( {x_{{{\text{best}}}} } \right)} \\ \end{array} } \right.$$
(30)

The flowchart of the proposed LINFO technique is displayed in Fig. 3. The place of Leader-based mutation-selection in the proposed algorithm is presented in this figure. This modification leads to enhance the exploration of the proposed LINFO algorithm based on the simultaneous crossover and mutation using the three best leaders.

Fig. 3
figure 3

Flowchart of proposed LINFO algorithm

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5 Simulation results

5.1 The performance of LINFO optimizer

The proposed LINFO technique's capability and proficiency are tested on the numerous benchmark functions, by the statistical measurement including the best values, worst values, median values, mean values, and standard deviation (STD) for optimal solutions reached by the proposed LINFO algorithm and the other recent and well-known optimization algorithms. The results reached with applying the proposed LINFO technique are compared to those obtained from four recent techniques including, gray wolf optimizer (GWO) [49], tunicate swarm algorithm (TSA) [50], butterfly optimization algorithm (BOA) [51], and the conventional INFO algorithm [47]. Figure 4 displays the Qualitative metrics on F1, F4, F5, F8, F9, F12, F15, and F18: 2D views of the functions, convergence curve, average fitness history, and search history.

Fig. 4
figure 4

Qualitative metrics of eight benchmark functions including: 2D image of the functions, search history, average fitness history, and convergence curve using LINFO technique

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Statistical analysis of the proposed LINFO technique and the other four optimization algorithms is presented in Tables 1, 2, and 3 when applied for three types of benchmark functions (unimodal, multimodal, and composite, respectively). The optimal values achieved by the LINFO, INFO, BOA, TSA, and GWO techniques are displayed in bold. It can be noted that the LINFO technique achieves the optimum solution for the majority of the tested benchmark functions. Figure 5 illustrates the convergence curves of all techniques for those functions. Whereas Fig. 6 depicts the boxplots for these techniques. It is clear from Figs. 5 and 6 that the proposed LINFO technique has achieved a stable point for all functions and that it is boxplots are stable and very narrow for the majority of functions when compared to those of the other algorithms.

Table 1 Unimodal results for benchmark functions
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Table 2 Multimodal results for benchmark functions
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Table 3 Composite results for benchmark functions
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Fig. 5
figure 5figure 5

The convergence characteristics of all optimization algorithms for 23 benchmark functions

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Fig. 6
figure 6figure 6

Boxplots for l optimization algorithms for 23 benchmark functions

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5.2 Real-world application

To validate the effectiveness of the proposed approach in solving EM problem, 2 MG test systems were simulated. The grid- connected MG system is as illustrated in Fig. 2,

  • MG test system 1

In this test system, the MG is comprised of three DE units, one WT unit, one PV generator, and three residential customers with DR. Table 4 includes the cost coefficients of the DE units, three customers data, and the daily energy interruptibility limits for each customer. The values of power interabtabilty at each hour for customers are assumed to be the same as listed in Table 5. Figure 7 shows the hourly output power of WT and PV generators [43]. It is assumed that the power transacted cost between the MG and the main grid has symmetrical values; MG operator's daily budget is $500 for this test system.

Table 4 DE units data and customer data (Test system 1)
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Table 5 λi,t values and total initial hourly demand (test system 1)
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Fig. 7
figure 7

Hourly solar and wind power for test system 1

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All simulations were conducted using MATLAB 2021b on a 2.9-GHz i7 PC with 8-GB of RAM. To address the EM problem and integrate the DRP for the purpose of cost minimization and benefit maximization, 20 separate runs were carried out using PSO [52], JAYA [53], HBA [54], original INFO [47] and the proposed INFO algorithms. The results obtained from these runs were compared, and the findings are presented in Table 6. Notably, it is evident that the LINFO technique outperforms the other techniques in terms of cost minimization.

Table 6 Comparison of the EM problem for test system 1
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The convergence characteristics curve of the objective function is shown is Fig. 8, which demonstrate the effectiveness and robustness of LINFO algorithm.

Fig. 8
figure 8

Convergence characteristic curve of test system 1

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Using LINFO, the hourly generated power from the three DEs and the power transacted with the grid are shown in Fig. 9; it can be noted that during the period of renewable sources generation, the power is mainly sold to the main grid from the MG with total power transaction of -18.69 kW (the power sold to main grid is higher than that bought from it). The initial load and load with DR are shown in Fig. 10; the power curtailed from all customers is shown Fig. 11, as well as the incentive they get with total power curtailment of 102 KW. Total power curtailment for each customer during the day is shown Table 7.

  • MG test system 2

Fig. 9
figure 9

Hourly DE units generated power and grid power for test system 1

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Fig. 10
figure 10

Load demand and final load with DR

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Fig. 11
figure 11

Power curtailed and incentive paid to customers for test system 1

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Table 7 Total amount of energy saved and the total amount of incentive received (Test system1)
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For scalability validation of the proposed algorithm, a second test system with larger MG is simulated; this test system the MG is comprising of 10 DE units, ten WT generators, ten PV generator and seven customers with DR. Tables 8, and 9 list the cost coefficients of the DE units and seven customers' data and the daily energy interruptibility limits for each customer, respectively. The initial load for this test system is shown in Fig. 12; Fig. 13 shows the hourly values of WT and PV generators [43]. The values of power interabtabilty are shown in Table 10.

Table 8 Ten conventional generators units data for test system 2
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Table 9 Customer type, coefficients of the customer cost function, and daily customer curtailment limitations (test system 2)
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Fig. 12
figure 12

Initial hourly load demand for test system 2

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Fig. 13
figure 13

Hourly solar and wind power for test system 2

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Table 10 Power interabtabilty (λi,t) values for test system 2
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For this test system, Fig. 14 shows the hourly generated power from the ten DEs and the power transacted with the utility grid Fig. 14; the initial load and load with DR are shown in Fig. 15, as it is shown the DRP help in decreasing the load demand mainly in the period where the renewable generation is low, Total power curtailment for each customer during the day is shown Table. 11.

Fig. 14
figure 14

Hourly DE units generated power and grid power for test system 2

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Fig. 15
figure 15

Load demand and final load with DR test system 2

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Table 11 Total amount of energy saved and the total amount of incentive received (Test system 2)
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6 Conclusion

In this paper, a LINFO algorithm has been proposed in order to solve the EM problems in MGs with DRP. The superiority of LINFO over the standard INFO algorithm, as well as other optimization techniques (e.g., BOA, TSA, and GWO) in terms of convergence curve, average fitness history, and search history is demonstrated using different bench functions. Also, a statistical analysis of the proposed LINFO technique and four optimization algorithms are compared for three unimodal, multimodal, and composite benchmarks, which demonstrate that LINFO has achieved the optimal solution for most of test functions. The proposed LINFO is used to operate the MG with minimum DEs generation cost and minimum cost of power transaction. Different optimization approaches were used to solve the energy management problem in MG over a period of 24 h in order to get optimal operation of MG sources. A comparison among the convergence characteristics of proposed LINFO algorithm and the original INFO algorithm prove the adequacy of the proposed LINFO in solving the EM problem with fast convergence. LINFO results are compared with well-known and new developed optimization techniques as PSO, JAYA, HBA, and INFO; the results demonstrated the robustness of the proposed algorithm to solve the EM problem in MG. to test system were simulated; in test system 101 KWh reduction in energy consumption, and 2677 MWh in the second test system. The results of two test systems for the EM problem under consideration have been described.

In addition, the proposed LINFO algorithm can be used to solve an probabilistic EM problem. Ref. [13] proposed a cloud based infrastructure for handling the data of smart buildings in Nano- Microgrid. The dependent on the main grid is reduced in the peak hours. The proposed LINFO algorithm with the demand response, which is an ancillary service in the smart grid, can be employed to solve the EM in the cloud based nano-grid.