Enhanced Aquila optimizer algorithm for global optimization and constrained engineering problems

1.   Introduction

Economic dispatch (ED) ^[1] in power systems is an important issue for obtaining the steady-state and economic operations of systems that is a typical constrained optimization problem with multiple variables. The optimization goal of the ED problem is to determine the most economic power outputs of generators while satisfying multiple constraints, such as the generation capacity limits, power demand balance, network transmission losses, ramp rate limits and prohibited operating zones. Considering the valve-point effects (VPE) of multivalve steam turbines for the ED problem, the objective cost function is a nonlinear and nonconvex function, which is hard to solve ^[2]. Especially in large-scale power systems with multiple generators, the ED problem is a complex optimization problem with several local optimal solutions, and thus the global optimal solution is hard to find.

In recent years, several optimization algorithms, including conventional algorithms and meta-heuristic algorithms, have been proposed to solve the ED problems. Some conventional algorithms, such as linear programming (LP) ^[3], self-adaptive dynamic programming (SADP) ^[4], iterative dynamic programming (IDP) ^[1] and evolutionary programming (EP) ^[5], have been applied to solve the ED problems. These methods solve the ED problems using the simplified optimization model in which the valve-point effects, ramp rate limits, prohibited operating zones and transmission losses are not considered. Moreover, the optimal results obtained by these methods may be the local optima and have lower computational accuracy. The drawbacks of conventional algorithms prompt researchers to study meta-heuristic algorithms for solving ED problems.

Recently, many meta-heuristic algorithms have been proposed to solve the various optimization problems, such as flow shop scheduling ^[6,7,8], steelmaking scheduling ^[9], job shop scheduling ^{[10,11,12,13]}, flexible task scheduling ^[14] and chiller loading optimization ^[15,16,17]. Due to the better optimization performance, many meta-heuristic algorithms have also been applied to solve the complex ED problems, and these algorithms include the genetic algorithm (GA) ^{[18,19,20,21]}, particle swarm optimization (PSO) and its variants ^{[22,23,24,25,26]}, firefly algorithm (FA) ^[27], oppositional real coded chemical reaction optimization (ORCCRO) ^[28], differential evolution (DE) ^[29,30], chaotic bat algorithm (CBA) ^[31], oppositional invasive weed optimization (OIWO) ^[32], teaching learning based optimization (TLBO) ^[33], tournament-based harmony search (THS) ^[34], grey wolf optimization (GWO) ^[35,36], hybrid artificial algae algorithm (HAAA) ^[37], orthogonal learning competitive swarm optimizer (OLCSO) ^[2], backtracking search algorithm (BSA) ^[38], social spider algorithm (SSA) ^[39], civilized swarm optimization (CSO) ^[40], kinetic gas molecule optimization (KGMO) ^[41] and hybrid methods ^{[42,43,44,45]}. Although the above meta-heuristic algorithms have been shown to be efficient in solving ED problems, the optimal results obtained by these algorithms are not the most economical.

By mimicking the colonization behavior of weeds in nature, the invasive weed optimization (IWO) algorithm was proposed by Mehrabian and Lucas ^[46] to optimize multidimensional functions. The experimental results demonstrated that IWO can obtain superior optimization results compared to other evolutionary-based algorithms. Due to its robustness, convergence, high accuracy and searching ability, the IWO algorithm has been applied to solve many engineering optimization problems. However, when IWO is used to solve the ED problem in large-scale power systems, the optimization power outputs of generators obtained by IWO consumes more generation costs compared to the reported methods in literature. To further improve the optimization performance of IWO in solving ED problems, especially ED problems in the large-scale power systems, inspired by the effective application of hybrid methods in solving ED problems ^{[37,42,43,44,45]}, a hybrid invasive weed optimization (HIWO) algorithm that hybridizes IWO with GA is developed in this study. The motivation behind choosing GA integrated with IWO is to get a better dispatch solution using the crossover operation between offspring weed and its parent weed to improve the local search ability of IWO, and executing the mutation operation on offspring weeds to increase the diversity of the population. The main contributions of this study are as follows: (1) the economic dispatch problem with various practical constraints is investigated by minimizing the total power generation cost; (2) the crossover and mutation operations of GA are proposed to improve the optimization performance of IWO; and (3) an effective repair method of handing constraints is investigated to repair the infeasible dispatch solutions.

The rest of this paper is organized as follows. Section 2 gives the mathematical formulation of the ED problem. Section 3 introduces a hybrid invasive weed optimization (HIWO) algorithm. Section 4 presents the application method of HIWO on ED problems. Section 5 shows the experimental results and analysis on six power systems with different scales. The conclusion is finally given in Section 6.

2.   Problem formulation

The ED problem in power systems is to find the optimal dispatch solution of the power outputs of generators, while the total power generation cost of the system is minimized and all the constraints are satisfied.

2.1.   Optimization objective

The optimization objective of the ED problem is to minimize the power generation cost (SC) consumed by N number of generators in the power system, as shown in Eq 1.

where P_i and C_i are the power output and generation cost of the ith generator, respectively.

For the ED problem neglecting valve-point effects, C_i is calculated by Eq 2. For the ED problem considering valve-point effects, Eq 3 is used to calculate C_i ^[2,32].

where a_i, b_i and c_i are the cost coefficients of the ith generator; e_i and f_i are valve-point coefficients of the ith generator; $P_i^{min}$ is the lower limit of P_i.

2.2.   Constraints

The feasible dispatch solutions of the ED problem should satisfy the following constraints.

2.2.1.   Generation capacity limits

The power output of each generator must be in the range specified by the minimum ($P_i^{min}$) and maximum ($P_i^{max}$) of the power output of the ith generator, as shown in Eq 4.

2.2.2.   Power demand balance

The power outputs of generators should satisfy the system power demand (PD). For the ED problem neglecting network transmission losses (PL), the power demand balance is expressed as Eq 5 ^[30]. For the ED problem considering PL, the power demand balance is expressed as Eq 6.

PL can be calculated using the power flow analysis method ^[47] or the B-coefficients method ^[48]. This study adopts the following B-coefficients method to calculate PL.

where B_ij, B_0i and B₀₀ represent the loss coefficients.

2.2.3.   Ramp rate limits

In the actual operation of the power system, to avoid the excessive stress on the boiler and combustion equipment, the change rate of the power output of each generating unit should be within the ramp rate limit, as shown in Eq 8.

where $P_i^0$ is the power output of the ith generator at the previous time interval. UR_i and DR_i represent the upper limits of ramp up and ramp down rate of the ith generator, respectively.

When taking into account both the generation capacity limits and ramp rate limits, the value range of P_i can be rewritten as Eq 9.

2.2.4.   Prohibited operating zones

Considering the operation limitations of machine components, the power outputs of some generators cannot lie in the prohibited zones, as shown in Eq 10.

where ${P_{i, k -}^u}$ and ${P_{i, k}^l}$ represent the upper and lower limits of the kth prohibited zone, respectively. np_i is the number of the prohibited zones of P_i.

3.   Hybrid invasive weed optimization algorithm

3.1.   IWO algorithm

IWO is a novel evolutionary computation algorithm based on weed swarm intelligence. By simulating the propagation and growth behaviors of weeds in nature, IWO searches for the optimal solution of the problem in the solution space. The calculation steps of IWO include initialization, reproduction, spatial dispersal and selection. The initial population with Nw_o weed individuals is randomly generated in the feasible solution space, in which each weed consisting of variables represents a feasible solution. Then, each weed W_j in the population reproduces seeds, and the seeds grow into offspring weeds through spatial dispersal. The amount (Ns_j) of seeds reproduced by W_j is calculated by using Eq 11.

where Fit_j is the fitness value of W_j; Fit_min and Fit_max are the minimum and maximum fitness values in the weed population, respectively; Ns_min and Ns_max are the minimum and maximum of the number of seeds, respectively.

The parent weeds with higher fitness values can reproduce more seeds, and they have more offspring weeds in the population. This reproduction strategy means that IWO can converge rapidly and reliably to the approximate optimal solution. Offspring weeds are randomly distributed around their parent weed according to a normal distribution with a standard deviation (σ_it). The calculation formula of σ_it is shown in Eq 12. Along with the increase of the iteration times, σ_it is gradually reduced from an initial value (σ_iv) to a final value (σ_fv), which makes the search range of IWO be gradually reduced. This strategy makes IWO have the whole space search capability in early iterations and high local convergence in later iterations. After all the seeds grow into weeds, the Nw_max weeds with higher fitness values are selected from all the weeds as the parent weeds of the next iteration. Through Iter_max times iterations, the weed with the highest fitness value is the optimal solution of the problem.

where m is the nonlinear modulation index, and Iter and Iter_max are the current number and maximum of iterations, respectively.

3.2.   Framework of HIWO

In the proposed HIWO algorithm, IWO is used to explore the solution space around parent weeds. After the seeds reproduced by parent weeds have grown into offspring weeds, the crossover and mutation operations of GA are performed on offspring weeds for improving the quality and diversity of solutions, which can improve the convergence speed and avoid the premature convergence of the algorithm.

3.2.1.   Pseudo code of HIWO

The execution flow of HIWO is represented by the pseudo code shown in Figure 1.

3.2.2.   Crossover operation

Each offspring weed (OW_{(j, q)}) (q = 1, 2, …, Ns_j ) crosses with its parent weed (W_j) to generate a new weed ($OW{'_{\left( {j, q} \right)}}$). For each variable P_i (i = 1, 2, …, N), calculate the generation cost ($COW_{\left( {j, q} \right)}^i$) consumed by the P_i of OW_{(j, q)} and the generation cost ($CW_j^i$) consumed by the P_i of W_j, , and then determine the value of the P_i of $OW{'_{\left( {j, q} \right)}}$ according to the following two cases.

(a) If $COW_{\left( {j, q} \right)}^i \le CW_j^i$, the P_i of $OW{'_{\left( {j, q} \right)}}$ is equal to that of OW_{(j, q)}.

(b) If $COW_{\left( {j, q} \right)}^i > CW_j^i$, the P_i of $OW{'_{\left( {j, q} \right)}}$ is equal to that of W_j.

After the new offspring weed ($OW{'_{\left( {j, q} \right)}}$) is generated by the crossover operation, set $O{W_{\left( {j, q} \right)}} = OW{'_{\left( {j, q} \right)}}$.

3.2.3.   Mutation operation

For each offspring weed (OW_{(j, q)}) (q = 1, 2, …, Ns_j ), randomly select X mutation points from N variables $\left( {{P_1}, {P_2}, \cdots , {\rm{}}{P_N}} \right)$, and then modify the P_i of each mutation point using a random number that is distributed around P_i according to a normal distribution with a standard deviation (${{\sigma _m}}$). The calculation formula of ${{\sigma _m}}$ is expressed as Eq 13.

where rand (0, 1) is a random number between 0 and 1.

4.   Application of HIWO on ED problem

4.1.   Implementation method

In the proposed HIWO algorithm, the first task is the encoding to represent each solution considering all of the constraints. Each weed (W_j) is represented as a row vector consisting of power outputs of generators, as shown in Eq 14. The weed population is initialized by randomly generating the power outputs of generators by using Eq 15. Then, infeasible weeds are repaired into feasible solutions by using the repair method in Section 4.2. Weeds in the initial population are used as the parent weeds to reproduce seeds, which grows into offspring weeds through spatial dispersal. The weeds with higher fitness value can reproduce more seeds. The fitness function used in this study is shown in Eq 16. Each offspring weed will perform the crossover and mutation procedures, like in the canonical GA, and thus can increase the diversity of the population. Then, the repair procedure is applied on the infeasible offspring weeds to make them satisfy with all of the constraints. If the total quantity of parent weeds and offspring weeds is larger than the specified population size, select the weeds with higher fitness values as the parent weeds of the next iteration. Otherwise, all the weeds are used as parent weeds. After multiple times iterations, the best weed with the highest fitness value is selected as the optimal dispatch solution of the ED problem.

where SC_j and Fit_j represent the power generation cost and fitness value of the jth weed, respectively.

4.2.   Repair method of infeasible solutions

An effective repair method of handing constraints is proposed in this study to repair infeasible weeds into feasible solutions. The detail repair steps are stated in the following.

Step 1: Modify the ${P_i}\left( {i = 1, 2, \cdots , N} \right)$ of W_j to satisfy the generation capacity limits and ramp rate limits by using Eq 17. If P_i violates the constraint of prohibited operating zone, update P_i using a upper or lower limit of the prohibited operating zones that is closest to P_i.

Step 2: Calculate the constraint violation (V) of the power demand balance. For the ED problem considering transmission losses, V is calculated by using Eq 18. For the ED problem neglecting transmission losses, V is calculated by using Eq 19. If $V \ne 0$, go to Step 3. Otherwise, go to Step 5.

Step 3: Determine the modification sequence of N generators. For each generator i (i = 1, 2, …, N), calculate the modification value $P_i^{'}$ of P_i using the power outputs of the other N-1 generators. Create the set R = $R = \left\{ {1, 2, \cdots , N} \right\} - \left\{ i \right\}$ to store the indexes of the other $N-1$ generators excluding the ith generator. For the ED problem neglecting transmission losses, Eq 5 is modified to Eq 20 to calculate $P_i^{'}$. For the ED problem considering transmission losses, Eq 6 can be expressed as a second-order equation in $P_i^{'}$ as shown in Eq 21. One root of the second-order equation is chosen as $P_i^{'}$, as shown in Eq 22. Then, modify $P_i^{'}$ using Eq 17 to satisfy the generation capacity limits.

For each generator i (i = 1, 2, …, N), assume that the ith generator is selected as the revised generator, and P_i is replaced by $P_i^{'}$. Calculate the cost change (CX_i) using Eq 23 and the constraint violation (PV_i) of the power demand balance using Eq 18 or 19. Then, calculate the percentage (PCV_i) according to CX_i and PV_i, as shown in Eq 24. Finally, create a set (S) to store the modification sequence of N variables, which is determined by the value of PCV_i sorted in ascending order.

Step 4: Modify the power output of each generator in turn according to the modification sequence stored in S until the power demand balance constraint is satisfied. When the ith ($i \in S$ generator is selected as the modified generator, the modified value ($P_i^{'}$) of its power output is calculated by using Eq 20 or 22. If all the generators are modified and the power demand balance constraint is still not satisfied, go to Step 3. Otherwise, go to Step 5.

Step 5: Output the modified weed (W_j).

5.   Experimental results and analysis

To validate the optimization ability of HIWO on ED problems with various practical constraints, six classical ED problems in the small, medium, large and very large-scale power systems were selected as the studied test cases. For each test case, the optimal dispatch results obtained by HIWO in 50 independent runs, including the minimum cost (SC_min), average cost (SC_avg), maximum cost (SC_max) and standard deviation of the costs (SC_std), are compared to those of algorithms reported in the literature. The best optimization performance among these algorithms is shown in boldface. The parameters of HIWO on six test systems are set as follows: the initial population size Nw_o = 30, maximum population size Nw_max = 50, minimum number of seeds Ns_min = 1, maximum number of seeds Ns_max = 5, nonlinear modulation index m = 5, initial standard deviation ${\sigma _{iv}} = 2$, final standard deviation ${\sigma _{fv}} = 0.0001$, maximum number of iterations Iter_max = 2000, and number of mutation points X = 1 in the 160-generator system and X = 3 in the other systems. HIWO is implemented by using the MATLAB (R2016a) environment on an Intel core i7-4790 CPU with 8.00 GB RAM personal computer.

5.1.   Small-scale test system

The 15-generator power system ^[2,24] considering transmission losses, ramp rate limits and prohibited operating zones is selected as the small-scale test system. The power load demand of the system is 2630 MW. In this test system study, the optimal power outputs of generators obtained by HIWO are shown in Table 1.The optimal dispatch results of HIWO are compared to those of OLCSO ^[2], WCA ^[49], ICS ^[50], FA ^[27], RTO ^[51], EMA ^[52] and IWO, as shown in Tables 2. Compared to other algorithms in terms of minimum, average, maximum and standard deviation of costs in 50 runs, the dispatch solution obtained by HIWO consumes the least cost.

5.2.   Medium-scale test syste

The 40-generator power system ^[32] considering valve-point effects and transmission losses is selected as the medium-scale test system. The power load demand of the system is 10500 MW. The optimal power outputs obtained by HIWO are shown in Table 3. The optimal dispatch results of HIWO are compared to those of ORCCRO ^[28], BBO ^[28], DE/BBO ^[28], SDE ^[29], OIWO ^[32], HAAA ^[37] and IWO, as shown in Tables 4. Compared to other algorithms in the literature, the proposed HIWO algorithm can obtain the cheapest dispatch solution in terms of minimum, average and maximum of costs in 50 runs.

5.3.   Large-scale test system

To verify the dispatch performance of HIWO on large-scale power systems with multiple local optimal solutions, two cases studies are performed to compare the optimization results of HIWO and other algorithms. The detail information of these two cases is shown as follows.

Case Ⅰ: The 80-generator power system ^[37] considering valve-point effects. The power load demand is 21000 MW.

Case Ⅱ: The 110-generator power system ^[20,32] neglecting valve-point effects and transmission losses. The power load demand is 15000 MW.

In the case Ⅰ study, the optimal dispatch solution obtained by HIWO is shown in Table 5. The comparison results of generation costs generated by HIWO, THS ^[34], CSO ^[40], HAAA ^[37], GWO ^[35] and IWO are summarized in Table 6. It can be found from Table 6 that HIWO can obtain the cheapest dispatch solution compared to other algorithms.

In the case Ⅱ study, the optimal dispatch solution obtained by HIWO is shown in Table 7. The generation cost generated by HIWO are compared to those of ORCCRO ^[28], BBO ^[28], DE/BBO ^[28], OIWO ^[32], OLCSO ^[2] and IWO, which are summarized in Table 8. Compared to other algorithms in terms of minimum, average, maximum and standard deviation of costs in 50 runs, the optimal dispatch solution obtained by HIWO generates the least generation cost.

5.4.   Very large-scale test system

To investigate the dispatch performance of HIWO on very large-scale power systems, the following two cases studies are performed for comparing the optimization results of HIWO and other algorithms.

Case Ⅰ: The 140-generator Korea power system ^[23,32] neglecting transmission losses. The 12 generators consider the valve point effects. The power load demand is 49342 MW.

Case Ⅱ: The 160-generator power system ^[32] considering valve-point effects. The power load demand is 43200 MW.

In the case Ⅰ study, the optimal dispatch solution obtained by HIWO is shown in Table 9. The optimal results of HIWO are compared to those of SDE ^[29], OIWO ^[32], HAAA ^[37], GWO ^[35], KGMO ^[41] and IWO, as shown in Table 10. The corrected optimal result of OIWO is shown in italics. Compared to other algorithms in terms of minimum, average, maximum and standard deviation of costs in 50 runs, HIWO can obtain the cheapest dispatch solution.

In the case Ⅱ study, the optimal dispatch solution obtained by HIWO is shown in Table 11. The optimal results of HIWO are compared to those of ORCCRO ^[28], BBO ^[28], DE/BBO ^[28], CBA ^[31], OIWO ^[32] and IWO, as shown in Table 12. Compared to other algorithms, HIWO can also obtain the cheapest dispatch solution in terms of minimum, average, maximum and standard deviation of costs.

5.5.   Convergence tests

To illustrate the convergence ability of HIWO for solving different-scale ED problems with various constraints, the convergence curves of HIWO and IWO on six test systems are drawn, as shown in Figure 2. It can be found from Figure 2 that HIWO can converge to the optimal areas in the six test systems, and the convergence speed of HIWO on the 15, 40, 80, 110 and 140-generator power systems, is faster than that of IWO. Although the convergence speed of HIWO on the 160-generator power system is slower than that of IWO in the early evolutionary stage, it is faster than that of IWO in the later evolutionary stage. The reason is that the crossover and mutation decrease the fitness value of offspring weeds in 160-generator power system having lots of constraints, and then reduce the convergence speed in the early evolutionary stage, but increase the diversity of the population to jump out local optimization in the later stage.

6.   Conclusion

In this paper, a hybrid HIWO algorithm combining IWO with GA is proposed to solve ED problems in power systems. The HIWO adopts IWO to explore the various regions in the solution space, while the crossover and mutation operations of GA are applied to improve the quality and diversity of solutions, thereby preventing the optimization from prematurity and enhancing the search capability. Moreover, an effective repair method is proposed to repair infeasible solutions to feasible solutions. The experimental results of the six test systems studies show that HIWO can obtain the cheapest dispatch solutions compared to other algorithms in the literature, and have a better optimization ability and faster convergence speed compared to the classical IWO. In summary, the proposed HIWO algorithm is an effective and promising approach for solving ED problems in different-scale power systems.

Acknowledgments

This research is partially supported by the National Science Foundation of China under grant numbers 61773192 and 61773246, the Key Laboratory of Computer Network and Information Integration (Southeast University), the Ministry of Education (K93-9-2017-02), and the State Key Laboratory of Synthetical Automation for Process Industries (PAL-N201602).

Conflicts of interest

The authors declare no conflict of interest.