Keywords

1 Introduction

DED is an extension of the issue of static economic transmission (SED). SED scenario finds the cost-efficient production combination of generators to fulfil the anticipated demand for a single load at a particular time hour. Because of the high-power system load fluctuation, SED could not meet the operating restrictions of the generators. The primary aim of the DED is to reduce the overall cost of production while meeting the limitations of equality, inequality and dynamic restrictions. Moreover, owing to look ahead inability, the outcomes of SED will be suboptimal when evaluating a time horizon moment of a time instance [1]. The balance of load demand is the constraint on equality, and the restrictions on the forbidden area and limitations of capacity generation are the constraints of inequality. The solution of the DED issue is more complex by considering these dynamic restrictions. Much work has been expended in trying to successfully address the essential but complex DED issue, and a variety of solution approaches have been suggested. Until now, these techniques have been experimentally divided into two groups: classical and heuristic methods. Classical methods include Lagrangian method [2], quadratic programing [3] and dynamic programing [4], etc., and while they offer some benefits like great calculation efficiency and theoretically optimal [5], they have several drawbacks as well. As a substitute for traditional methods, heuristic techniques have received much attention and proven their efficacy as strong optimizers for the issue of DED in the past several decades, like evolutionary programing [6] particle swarm optimization (PSO) [7], differential evolution (DE) [8], artificial bee colony (ABC) [9], krill herd algorithm (KHA) [10] and artificial immune system (AIS) [11].

In 1960, John Holland invented the genetic algorithm (GA) [12]. To date GA has been used to resolve a number of real-world issues of optimization [13,14,15]. It may quickly reach the global minimum search area, and it takes more time to converge. A hybrid approach is one way of tackling this problem. Several GA variations were thus presented to avoid the disadvantage trap in local optima and reach global solution with in less time [16]. The major contributions in this paper are as follows:

  1. (i)

    Consider DED problem instead of classical ED, since introduction of dynamic constraints makes the DED problem more complicated.

  2. (ii)

    DED problem is solved using newly created a variant of GA with three-parent crossover. This method introduced a three-parent crossover and a typical mutation via a diversity operator, resulting in maintain efficient chromosomes.

The effectiveness of GA-TPC is shown with two distinct test systems. The remaining paper is arranged as follows, Sect. 2 provides a mathematical model of DED problem considering valve points, Sect. 3 offers about GA-TPC algorithm, Sect. 4 shows three different cases, and achieved results compare with the outcomes of the latest techniques and the final conclusion in Sect. 5.

2 Mathematical Model

DED is required to optimize the overall cost of all thermal generators exposed to different restrictions on a regular basis over a time horizon. The thermal cost characteristics, associated constraints and basic formulations are discussed more below [7].

2.1 Optimization of Total Cost (TC)

Usually, the DED problem's goal function may be approximated by a simple quadratic equation [7].

$$\min \,f = \sum_{t = 1}^T {\sum_{m = 1}^{{\text{NG}}} {a_m + b_m P_{{\text{Gm}},t} + C_m P_{{\text{Gm}},t}^2 } }$$
(1)

where f gives TC of all generators; PGm,t indicates active power of mth generator at tth hour.

2.2 Optimization of TC with Valve Points (TCV)

However, the production curve for multi-valve steam units differs considerably in comparison with the quadratic function of the active power output. The inclusion of a valve point effect on the fuel cost of the producing unit provides a better representation of the cost of fuel. As the valve point is completed with spiking, the fuel price function includes more nonlinear series. A non-convex function to assess the effect of the valve points is thus employed in the study given below [7].

$$\min \,f = \sum_{t = 1}^T {\sum_{m = 1}^{{\text{NG}}} {a_m + b_m P_{Gm,t} + C_m P_{Gm,t}^2 } } + \left| {d_m \times \sin (e_m (P_{{\text{Gm}},t}^{\min } - P_{{\text{Gm}},t} ))} \right|$$
(2)

where \(a_m ,b_m ,c_m ,d_m \& e_m\) indicate cost coefficients of \(m{\text{th}}\) generator.

2.3 Constraints

The limitations in the current work are briefly described below [7].

Equality constraints: It is a real power balance constraint and is given below,

$$\sum_{n = 1}^{{\text{NG}}} {P_{Gnt} = P_D (t)} + P_{{\text{loss}}} (t)\,\,\,\,\,\,\,\,\,\,t = 1,2,...T$$
(3)

where \(P_D\) reports load demand, and \(P_{{\text{loss}}}\) indicates transmission loss and is calculated as follows,

$$\sum_{k = 1}^{{\text{NG}}} {\sum_{m = 1}^{{\text{NG}}} {P_{kt} B_{km} P_{mt} } } + \sum_{k = 1}^{{\text{NG}}} {B_0 P_{kt} + B_{00} } \,\,\,\,\,\,\,\,\,\,\,\,\,t = 1,2,...T$$
(4)

where \(B_{km} ,B_k \& B_{00}\) are called loss coefficients.

Inequality constraints: These are expressed among their low and high limits and are given below,

$$P_{Gn}^{{\text{min}}} \le P_{Gnt} \le P_{Gn}^{{\text{max}}} \,\,\,\,\,\,\,n = 1,2, \ldots ,N_G \,\,\,\,\,\,\,\,\,\,\,\,\,\,t = 1,2, \ldots ,T$$
(5)

3 Proposed Genetic Algorithm with Three-Parent Crossover

Different GAs for many real-world numerical problems have been presented over several decades. However, the effectiveness of the various approaches is dependent only on features of the objective function. In certain instances, GA did not perform nor was compared with other algorithms [17, 18]. Therefore, GA performance is improved by adding three-parent crossover instead of a typical two-point crossover, and diversity operator is applied instead of a fairly regular mutation [17]. The current crossover uses three parents to produce three new children, helping explore and leverage the diversity operator.

Crossover is a GA operator of great importance. It is responsible for recombination structure and GA convergence speed. The conventional GA combines the chromosomes from the two chosen parents to produce a new chromosome which inherits information regions contained in parent chromosomes. The crossover suggested in the GA-MPC is based on an idea of heuristic crossover, and here, a child (c) is created from a set of two parents (a, b), like \(c = a + rand(a - b)\), where ‘rand’ is a random number among 0,1. The GA-MPC nevertheless uses three rather than two parents.

The procedure for the proposed algorithm is explained below.

  1. (i)

    Selection

Selection of the parents is a simple process by which parents are chosen based on fitness of the chromosomes. The likelihood of adding additional offspring to the following generation is that solutions with high fitness ratings. A basic selection of roulette wheels rule utilized in our approach [19].

  1. (ii)

    Proposed three-parent crossover

Crossover procedure is very important in GA. To generate new offspring, the crossover must be able to use search space information. Offspring distribution should neither be disproportionately narrow or disproportionately large compared to that of their parents. It is possible that the offspring will lose diversity and converge early if their distribution is much smaller than that of their parents. The opposite may be true if the children are dispersed extensively, in which case they may be too varied and require an excessively long time to converge to optimality. There should be a balance between exploration and exploitation in the next generation. Based on the aforementioned idea, in the proposed work, three parent crossover based on random procedure is used rather than regular two parent crossover. The procedure is given below [17].

  1. 1.

    Select the parent individuals by using selection process.

  2. 2.

    If any two individuals are similar, then one is replaced with randomly from selection pool.

  3. 3.

    Arrange those three individuals according to best to worst fitness value.

  4. 4.

    A number ‘Ɛ’ is produced randomly;

    1. (a)

      New off springs are produced by using following equations

      $$\begin{gathered} {\text{OF}}_1 = x_1 + \varepsilon (x_2 - x_3 ) \hfill \\ {\text{OF}}_2 = x_2 + \varepsilon (x_3 - x_1 ) \hfill \\ {\text{OF}}_3 = x_3 + \varepsilon (x_1 - x_2 ) \hfill \\ \end{gathered}$$
      (6)

where x1, x2 & x3 are the selected parents by using selection process, and OF1, OF2 & OF3 denote newly generated off springs.

  1. (iii)

    Diversity operator

To improve the exploitation capability in the individuals, diversity operator introduced in [14] considered here.

The step-wise procedure of GA-TPC to solve ED is given below:

Step 1: Initialize GA-TPC variables, max generations (Gmax).

Step 2: Each chromosome in GA-TPC is a solution to a DED issue. The kth chromosome in mth generation is expressed in below given form

$$\begin{gathered} X_k^m = \left[ {\begin{array}{*{20}c} {P_{g1,1,k}^m } & {P_{g1,2,k}^m } & \cdots & {P_{g1,t,k}^g } \\ {P_{g2,1,k}^m } & {P_{g2,2,k}^m } & \cdots & {P_{g2,t,k}^m } \\ \vdots & \vdots & \ddots & \vdots \\ {P_{gNg,1,k}^m } & {P_{gNg,2,k}^m } & \cdots & {P_{gNg,t,k}^m } \\ \end{array} } \right]\begin{array}{*{20}c} {k = 1,2 \ldots {\text{NP}}} \\ {g = 1,2 \ldots G_{{\text{max}}} } \\ \end{array} \hfill \\ \hfill \\ \end{gathered}$$
(7)

where \(t\) indicates number of intervals in the dispatch period.

Step 2: Evaluate fitness of every individual using Eq. 8.

$$\left| F \right| = f + w_P \left( {|P_{G1} - P_{G1}^{\lim } |} \right)^2$$
(8)

where \(w_{P}\) indicates penalty value of slack bus real power.

Step 3: Apply the selection, proposed crossover, diversity operator, and create new generation.

Step 4: If any variable exceeds its existing limits, then it will be set to inline high or low value.

Step 5: Terminate the process, if utmost iterations are marked, and take the best result from previous iteration as best solution. Else, go to Step 2.

4 Simulation Results

Two different modules are investigated to assess the feasibility and efficacy of the GA-TPC technique suggested in the solution of the DED issue. The dispatch time is chosen as 24 h for one day. The number individuals and utmost iterations in all the cases are considered 40 and 300, respectively. The following are the two cases:

M1: a three-generator system without point loadings.

M2: a ten-generator system with valve point loadings.

4.1 M1: 3 Unit System

The proposed system consists of three generators and complete data for this system that includes cost characteristics of generators, generator limits, and load demand in each interval is referred from reference [20]. The optimal set of active powers obtained to this system with GA & GA-TPC are given in Table 1. These results are compared with CSA [20] and ISA [20], RGM [21] and ACO [21] and are given in Table 2. From this table, it is noticed that the suggested approach provides a superior way to discover solutions to such complex DED issues, with minimum, average and maximum costs. A minimum of 176,017.5363 ($/day) and a minimum of 176,059.3264 $/day achieved utilizing the formulations of proposed GA-TPC and original GA, showing the remarkable nature of the suggested method. In addition, the convergence characteristic of the method suggested is compared and shown in Fig. 1 with the original GA. This figure indicates that both convergence speed and optimum objective function of the proposed GA-TPC beats conventional GA. Here, 3-unit system size is very small, so the deviation of optimal cost from GA to GA-TPC is very small. Thus, the convergence curves are much closer to each other.

Table 1 Optimal solution of 3-unit system without VPL using GA & GA-TPC
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Table 2 Comparison of the statistical analysis for 3-unit system with the other methods
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Fig. 1
figure 1

Convergence curve of 3-unit system

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4.2 M2: 10 Unit System

GA-TPC performance is identified by considering 10 unit systems for solving DED problem with inclusion of valve points. This system is believed to have a complete date from [7]. Table 3 illustrates the findings achieved for the 10-unit system with valve point loading effect. These findings are compared to those of previously developed algorithms such as DE [8], hybrid EP-SQP [6], hybrid PSO-SQP [18], deterministically guided PSO (DGPSO) [7], hybrid DE (HDE) [7], improved DE (IDE) [7], ABC [6], modified DE (MDE) [7], AIS [12], AIS-SQP [12], chaotic DE (CDE) [7] and improved PSO (IPSO) [7]. This table shows a comprehensive comparison of solution quality, including lowest, average and maximum cost, as well as simulation time, and it is confirmed that the proposed method produces more optimum results outcomes that the methods described in the literature. Tables 4 and 5 shows the optimal set of active powers obtained to this system with GA-TPC and GA respectively. The suggested algorithm's convergence characteristic is shown in Fig. 2 and compared to the original GA. As can be seen from the graph, the suggested method beats the original GA in terms of convergence speed and optimality. The variation of TC with 20 trials is shown in Fig. 3 for 6-unit system, and it is observed that 17 trials were achieved optimal cost by the GA-TPC method over 20 trials and indicates the precision of the proposed method. Aforementioned simulation results depict that GA-TPC is successful in addressing small-scale test systems and using it to solve multi-objective DED for large and practical power systems would be an extension of the current study.

Table 3 Comparison of the statistical analysis for 10-unit system with the other methods
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Table 4 Optimal solution of 10-unit system using GA-TPC
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Fig. 2
figure 2

Convergence characteristics of 10-unit system

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Fig. 3
figure 3

Variation of TCV for 10-unit system with 20 trials

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Table 5 Optimal solution of 10-unit system using the GA
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5 Conclusion

To address the dynamic economic dispatch issue of power systems with valve point loading effects, this article proposes a novel method termed genetic algorithm with three-parent crossover. Two different test scenarios are used to validate the technique. Comparing the suggested technique to other previously published approaches, including lowest, average and maximum costs as well as simulation time, provides a thorough understanding of the pros and cons of each. The findings of the study show that GA-TPC was able to find solutions that were more cost-effective. The comparison of suggested algorithm's convergence characteristics with conventional GA also confirms the speed and ability of the GA-TPC method to discover superior solutions. These facts suggest that the technique under consideration is capable of resolving DED problems.