Abstract
Load flow analysis is a significant tool for proper planning, operation, and dynamic analysis of a power system that provides the steady-state values of voltage magnitudes and angles at the fundamental frequency. However, due to the absence of a slack bus in an islanded microgrid, modified load flow algorithms should be adopted considering the system frequency as one of the solution variables. This paper proposes the application of nature-inspired hybrid optimization algorithms for solving the load flow problem of islanded microgrids. Several nature-inspired algorithms such as genetic algorithm (GA), differential evolution (DE), flower pollination algorithm (FPA), and grasshopper optimization algorithm (GOA) are separately merged with imperialistic competitive algorithm (ICA) to form four hybrid algorithms named as ICGA, ICDE, ICFPA, and ICGOA. Performances of these algorithms are tested on the 6-bus test system and the modified IEEE 37-bus test system. A comparison among the proposed algorithms is carried out in terms of statistical analysis conducted using SPSS statistics software. From the statistical analysis, it is identified that on an average, ICDE takes less number of iterations and consequently needs less execution time compared with other algorithms in solving the load flow problem of islanded microgrids.
IN modern time, microgrid systems have evolved as an organized and flexible architecture comprising of distributed energy resources (DERs), which can be a potential replacement of the aging electrical infrastructure with enhanced operability and reliability, and reduced CO2 emission to alleviate the environmental change. Due to its functionality as an aggregated distributed generation (DG) unit in both islanded and grid-connected modes, microgrid has gained much acceptance over the years [
In [
In most of these studies, the system model was developed in the stationary reference frame considering the voltages and currents as phasors, which only allowed steady-state analysis of the system. However, these studies lacked necessary information for linearizing a dynamic model of the system. There are angular differences between synchronous reference frames of the generation units, which are not determined by these methods. For consistent system-level operation points, these angle differences are critical to obtain. In [
Nature-inspired optimization algorithms can be good alternatives to the gradient-based techniques in obtaining a global solution. Multi-solution-based evolutionary algorithms and swarm-intelligence-based algorithms have a better possibility of avoiding a local optimum by exploring a larger portion of the search space [
In this paper, the application of hybrid nature-inspired optimization algorithms was proposed to solve the load flow problem of droop-controlled islanded microgrids in the synchronous reference frame. To come up with a fairly optimal result, several multi-solution-based algorithms such as GA, differential evolution (DE) algorithm, flower pollination algorithm (FPA) and grasshopper optimization algorithm (GOA) were adopted, and each of them was separately combined with ICA to obtain four hybrid algorithms, namely ICGA, ICDE, ICFPA, and ICGOA. Statistical analysis was conducted to compare the performance of each of the hybrid algorithms in obtaining the optimal solution.
The rest of the paper is organized as follows. Section II contains a brief discussion on the mathematical model of the droop-controlled microgrids based on different literatures. In Section III, discussions regarding the load flow approach and the proposed hybrid algorithms are presented. Comparison among the proposed hybrid algorithms, along with the load flow results obtained for two case study systems, is carried out in Section IV. Finally, the conclusions are drawn in Section V.
Multiple DERs are aggregated in a microgrid system. Due to the nature of energy produced, in most cases, it is not suitable to directly connect these DERs to the distribution network. Thus, to convert the energy produced by a DER to the desired form, power electronic inverters are associated with these DERs before connecting to a distribution bus. As a result, developing the mathematical model of the inverter, along with its associated controllers, is important for the analysis of microgrid systems. The control strategy of an inverter coupled with an individual DER, is shown in

Fig. 1 Block diagram of control strategy of droop-controlled inverter for individual DER.
The three-phase output voltage across the filter capacitor , the three-phase output current through coupling inductor , and the three-phase current through the filter inductor are transformed to synchronous reference frame through abc to dq transformation. As shown in
In the conventional load flow analysis, the voltage and frequency of the slack bus are constant. However, in the case of islanded microgrid, the concept of the slack bus is not applicable as the system frequency is variable. As a result, for an islanded microgrid with the droop-controlled inverter, the system frequency has to be considered as one of the load flow variables along with the voltage magnitudes and reference angles contributed by each inverter in the system. The state variable can be described in terms of the load flow variables as:
(1) |
where is the system frequency; () is the angular difference between the local reference frame and the global reference frame for the
(2) |
(3) |
(4) |
The objective of the load flow analysis is to minimize the square of the absolute summation of errors in the active and reactive power mismatches of the inverters. The objective function can be written as:
(5) |
where and are the active and reactive power mismatches, respectively.
For a droop-controlled inverter, the power mismatch equations indicate the difference between the output power of the inverter calculated at the global reference frame and the reference values set by the droop controllers. For the inverter, the active and reactive power mismatch equations are:
(6) |
(7) |
where and are the d- and q-axis components of voltage magnitude across the filter capacitor branch of inverter transferred in the global reference frame, respectively; and are the d- and q-axis components of current magnitude through the coupling inductor of the inverter bus transferred in the global reference frame, respectively; and are the nominal values of system frequency and bus voltage, respectively; and and are the coefficients of the droop controller associated with the inverter.
To determine the power mismatch values, a set of equations has to be solved, which includes the calculation of bus voltages and output currents of inverters. A simplified equivalent circuit is given for this purpose, considering the output voltage of inverter across the filter capacitor as a voltage source behind its coupling impedance, as shown in

Fig. 2 Steady-state equivalent circuit of inverter model at bus i.
Step 1: following (S27) and (S28) in the supplementary material, the d- and q-axis components of the output voltage of the inverter are transformed in the global reference frame using the reference angle . Then, the output voltage of the inverter in terms of a complex quantity can be calculated as:
(8) |
(9) |
(10) |
where and are the d- and q-axis components of the voltage of inverter in its local reference frame, respectively.
Step 2: the current injected by the inverters can be easily calculated by transforming the circuit shown in
(11) |

Fig. 3 Norton equivalent circuit of steady-state inverter model at bus i.
where is the Norton equivalent current source; and is the coupling impedance as shown in
Step 3: the bus voltages can be calculated from the injected currents as shown in (12).
(12) |
where is the bus impedance matrix; and and are the vectors of voltages and injected currents at different buses of an islanded microgrid, respectively. For an N-bus system, , and will have the dimensions of , and , respectively. For islanded microgrids, the bus impedance matrix is a function of frequency, and it has to be updated at each iteration. For an N-bus system, the injected current at each bus () is given by:
(13) |
Step 4: after determining the bus voltages, the output current of the inverter can be determined by (14).
(14) |
The d- and q-axis components of the output current of the inverter in the global reference frame is given by:
(15) |
(16) |
Step 5: the equations from (8) to (16) are sufficient to calculate the mismatch values of active and reactive power for each inverter by solving (6) and (7).
Thus, these values of the power mismatch equations can be used to evaluate the objective function, as indicated in (5).
The flowchart of the hybrid algorithms employed in this study is given in

Fig. 4 Flowchart of algorithms used for load flow analysis.
Four other metaheuristic algorithms, GA, DE, FPA, and GOA, were separately combined with ICA to obtain four hybrid algorithms, namely ICGA, ICDE, ICFPA, and ICGOA. The detailed explanations for GA, DE, FPA, and GOA can be found in [
Step 1: initialize the system data of islanded microgrids.
Step 2: generate the initial population for the state variables in (1). In the case of ICA, the population individuals are called countries. For this study, the total number of countries in the population is set to 100.
Step 3: for each country, (8) to (16) are solved, and the mismatch values of active and reactive power for each inverter are calculated using (6) and (7). Then, for each country in the population, the value of the objective function is determined using (5).
Step 4: the countries are sorted according to their objective function values. Then, depending on the fitness values of the countries, the empires are generated by setting a specific number of countries as the imperialists and assigning the rest of the countries as colonies to them. For this study, among the 100 countries, 5 are chosen as imperialists, and the rest are assigned as colonies to the imperialists.
Step 5: the positions of the colonies are then moved towards the position of the imperialist by a process called assimilation. How much a particular colony will move towards the imperialist depends upon the assimilation co-efficient and the distance between the colony and the imperialist.
Step 6: the positions of some of the colonies are perturbed randomly by performing an operation called revolution. First, whether a particular colony will undergo revolution or not is determined through the probability of revolution, and then the perturbation is performed randomly based on the revolution rate .
Step 7: if there is a colony that has a lower fitness value than the imperialist, its position is interchanged. This process is referred to as intra-empire competition.
Step 8: the hybridization process of ICA with GA, DE, FPA, and GOA is carried out. Four distinct hybrid algorithms are obtained by following the four cases, as indicated in
Case 1: ICGA. GA is one of the most popular evolutionary optimization algorithms which was inspired by Charles Darwin’s theory of natural evolution. The real-coded version of GA is employed in this study. In GA, the initial solution sets are termed as parent chromosomes. For this study, the updated empires from the previous step are set as the parent chromosomes of GA. In order to generate offspring from the parent chromosomes, the extended-line crossover is performed for each child, two parents are randomly selected, and the crossover operation is carried out in terms of the extension rate for crossover . The generated offspring undergoes the mutation process to add diversity to the population individuals. A gaussian mutation operation is performed with respect to the mutation rate in order to generate the mutants. The number of offspring and the number of mutants to be generated in each iteration is determined in terms of the percentage of crossover and the percentage of mutation , respectively. Then, through a selection process, the fittest solutions are chosen for the next steps [
Case 2: ICDE. DE is a real-parameter optimization algorithm that also falls into the category of evolutionary algorithms. Here, the empires are assigned as parents of the DE algorithm. For a particular parent vector in the population, a differential mutation process is performed by randomly selecting three other distinct solution vectors. Then, a scaled difference is taken between any two of these three vectors in terms of a scaling factor , and the scaled difference is added with the third vector to obtain the mutant vector. Finally, the offspring is generated from the mutant and parent vector by exchanging components of the parent and mutant vector based on crossover probability . If the fitness of the offspring is better than the parent vector, their positions are interchanged. In this way, the positions of the countries (imperialists and colonies) are updated through the mutation, crossover, and selection process of DE [
Case 3: ICFPA. FPA is another evolutionary algorithm inspired by the pollination process of flowers. In this case, the empires are assigned as the population of flowers of FPA, and the positions of the population individuals are then modified by mimicking either global or local pollination process depending upon a probability switch . By imitating the concept of global pollination, a global searching process is introduced where Levy distribution is typically used to indicate the jump or fly distance of pollinators. On the other hand, the concept of local pollination is utilized in developing a local search operator which plays a vital role in exploiting the search area in the vicinity of the current solution [
Case 4: ICGOA. GOA is a swarm intelligence based algorithm which was proposed by mathematically modeling the swarming behavior of grasshoppers in nature. This mathematical model includes the model for the social interaction between grasshoppers. The attractive and repulsive forces between two grasshoppers are simulated through the social interaction function. Based on the intensity of attraction and the attractive length scale , the social interaction function is calculated, which is utilized in updating the positions of grasshoppers. Apart from that, the best solution up to the current iteration is considered as the target solution, which simulates the tendency of grasshoppers to move towards the source of food. While updating the positions of grasshoppers in each iteration, a deceleration coefficient is introduced to gradually obtain a balance between the exploration and the exploitation while chasing the target solution [
Selecting any one of the four cases in Step 8 is the principal difference among the four hybrid algorithms. The rest of the steps are similar for each algorithm.
Step 9: intra-empire competition is performed again, as mentioned in Step 7.
Step 10: first of all, the total cost of each empire is calculated in terms of the cost of the imperialist, the mean cost of the colonies, and their corresponding mean cost co-efficient . Then, in imperialistic competition, the weakest colonies are identified and given to the empires which have the most likelihood to possess them. If an empire ends up with no colonies, it will be eliminated.
Step 11: the solution set, which provides the best fitness value, is identified.
Step 12: if the stopping criteria are satisfied, the whole process will be terminated. Otherwise, the calculations will be repeated from Step 5. For this study, the optimization process will be terminated if one of the following criteria is satisfied.
1) The value of the best fitness is less than a pre-specified threshold value , which is set to be .
2) The total number of iteration is equal to a pre-specified value of the maximum number of iterations. For this study, the maximum number of iteration is set to be 50.
The values of the different parameters of these hybrid algorithms used for the simulations conducted in this study are summarized in Appendix A Table AI.
In Section II, the dynamic model of the microgrid system is discussed, and the proposed hybrid algorithms are outlined in Section III. In order to validate the applicability of these algorithms and to make a comparative analysis of these algorithms, the 6-bus test system and the modified IEEE 37-bus system are considered as the case study systems for load flow analysis. For both the systems, the simulations are performed using a personal computer with a processor of Intel Core i7-8550 of 1.8 GHz and with an installed RAM of 8 GB.
The single-line diagram of the 6-bus test system is shown in

Fig. 5 Single-line diagram of 6-bus test system.
Then, the load flow analysis is performed by applying ICGA, ICDE, ICFPA, and ICGOA to this case study system separately. These algorithms follow a stochastic process, and due to the inherent randomness of these algorithms, it is most likely that the number of iterations and the execution time needed to complete the optimization process may vary for each independent run. Thus, each algorithm is executed for 30 independent runs to make an overall comparison among the algorithms. For each run, the number of iterations to reach the stopping criterion and the overall execution time are recorded. These data are summarized in
The standard IEEE 37-bus system is modified by connecting seven inverters at different bus locations, as indicated in [

Fig. 6 Single-line diagram of modified IEEE 37-bus system.
Only constant impedance loads are considered in this case study. The branch and load parameters are considered to be the same as [
As done for the 6-bus test system, each of the four hybrid algorithms are applied for 30 independent runs. For each independent run, the number of iterations and the overall execution time to complete the load flow analysis are recorded. These data are summarized in
To further support the above discussion, the convergence graphs of each algorithm considering their best and worst results are shown in

Fig. 7 Convergence graphs for the best results of each algorithm. (a) Original version. (b) Enlarged version.

Fig. 8 Convergence graphs for the worst results of each algorithm. (a) Original version. (b) Enlarged version.
From
For further analysis of the acquired results, statistics software SPSS is used to perform statistical analysis of the obtained data from 30 independent runs. To demonstrate the uniqueness of each algorithm, the independent samples t-test is performed to compare the means of the data obtained from each algorithm. In this study, the data from two algorithms are defined as grouping variables at a time. Whenever independent t-test samples are performed in SPSS, the software generates corresponding F-test results which determine whether the data samples of two groups have equal variances or not.
For the F-test, if the p value is higher than the significance level of 0.05, the group variances are considered to be equal. Otherwise, equal variances can not be assumed. For the t-test, the null hypothesis H0 assumes that the mean values of the data sets are equal, and the alternative hypothesis H1 assumes that the mean values of the data sets are not equal. Whether the null hypothesis can be accepted or not depends on the p value of the t-test. From
The results of the load flow analysis of the two case study systems using ICDE are presented in
Among the 30 independent runs, the best results are tabulated here. In [
From
In this paper, the application of nature-inspired hybrid optimization algorithms is demonstrated for the efficient solution of the load flow problem of islanded microgrids. For solving the load flow problem, an objective function is formulated based on the square of the absolute summation of errors in the real and reactive power generations from the inverter-based microgrid sources. Using hybrid optimization techniques, namely ICGA, ICDE, ICFPA, and ICGOA, the objective function is solved for minimization. The hybridization is performed with a view to improving the global searching capability by an enhanced exploration of the search space. The 6-bus test system and the modified IEEE 37-bus system are considered to conduct the load flow analysis. The performances of the aforementioned hybrid algorithms are compared through a series of statistical tests. Based on the statistical tests, the ICDE is found to exhibit better performance than the other algorithms in terms of the required number of iterations and the execution time. Therefore, ICDE can be regarded as a prospective alternative to the conventional load flow techniques in the case of islanded microgrids.
Possible future research scope could be the consideration of the uncertainties of renewable energy resources and loads where the probabilistic models for the source and load have to be incorporated. Considering the load flow problem as a multi-objective optimization problem could be another possible future research direction where the power mismatch equations at each bus can be considered as separate objective functions.
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