Improved Parallel Differential Evolution Algorithm with Small Population for Multi-Period Optimal Dispatch Problem of Microgrids

Abstract

Microgrids have drawn attention due to their helpfulness in the development of renewable energy. It is necessary to make an optimal power dispatch scheme for each micro-source in a microgrid in order to make the best use of fluctuating and unpredictable renewable energy. However, the computational time of solving the optimal dispatch problem increases greatly when the grid’s structure is more complex. An improved parallel differential evolution (PDE) approach based on a message-passing interface (MPI) is proposed, aiming at the solution of the optimal dispatch problem of a microgrid (MG), reducing the consumed time effectively but not destroying the quality of the obtained solution. In the new approach, the main population of the parallel algorithm is divided into several small populations, and each performs the original operators of a differential evolution algorithm, i.e., mutation, crossover, and selection, in different processes concurrently. The gather and scatter operations are employed after several iterations to enhance population diversity. Some improvements on mutation, adaptive parameters, and the introduction of migration operation are also proposed in the approach. Two test systems are employed to verify and evaluate the proposed approach, and the comparisons with traditional differential evolution are also reported. The results show that the proposed PDE algorithm can reduce the consumed time on the premise of obtaining no worse solutions.

1. Introduction

A microgrid [1] is a low-generating and low-distributing power system composed of distributed generators, loads, energy storage systems, electric vehicles, protection devices, and control devices. Generally, renewable energy, such as solar and wind energy, is the main generating unit in a microgrid [2]. However, the output from solar and wind energy sources tends to fluctuate [3] and is often intermittent [4]. To make better use of these unstable renewable energy units, the optimal power dispatch of a microgrid is necessary [5].

The optimal power dispatch of a microgrid involves coordinating the power generation from sources such as wind turbines, PV systems, and micro-turbines, along with the charging/discharging cycles of energy storage systems and electric vehicle batteries, to achieve objectives such as minimizing operational costs [6] or optimizing economic and environmental impacts [7]. Through the optimal dispatch to a microgrid, surplus solar and wind energy are stored in the batteries, which may have prospects in smoothing and stabilizing the power output of photovoltaic systems, peak load shifting, operating reserves, and improving power quality [8]. The energy stored in batteries is released when solar and wind energy is deficient, such as at night, on cloudy days, and in a high-load period. The dispatch of a microgrid contributes to meeting the demand of loads in 24 h and makes better use of renewable energy [9].

The optimal power dispatch problem of a microgrid is a complicated constrained optimization problem and requires effective solution techniques. Traditional programming techniques such as MILP [10] and stochastic programming [11] have been proposed as alternative techniques to solve the optimization problem of the optimal dispatch of a microgrid. However, the optimal power dispatch problems in microgrids are generally non-smooth and non-convex optimization problems that are usually hard to solve through traditional optimization algorithms [12]. Artificial intelligence optimization algorithms have the ability to solve non-convex and non-smooth problems. Up to now, GA [13], GPSO-GM [14], modified PSO [15], harmony search algorithm [16], modified krill herd (MKH) algorithm [17], improved quantum PSO [18], SFO [19], DE algorithm [20], marine predator algorithm [21], quantum adaptive sparrow search algorithm [22], etc., have been employed to solve the optimal dispatch problem of microgrids successfully.

Among those artificial intelligence methods, the DE algorithm, consisting of the operations of mutations, crossovers, and selections, requires few control variables, is robust and easy to use, and converges fast [23] for a non-convex and non-smooth problem. Benefitting from the above advantages, many DE algorithms have been employed to solve different optimal power dispatch problems in microgrids. Research in Ref. [24] reformulated the nonlinear AC-DC OPF problem as an equivalent traditional AC OPF problem, and then a modified adaptive differential evolution algorithm was proposed to solve the non-convex AC OPF problem. Three soft computing techniques, viz. particle swarm optimization, differential evolution, and differential evolution with local–global neighborhoods, were used to perform a novel dynamic cost analysis of a microgrid and minimize its overall cost in Ref. [25]. In order to solve the real-time scheduling problem of community MG, a heuristic-based differential evolution (DE) algorithm was introduced in Ref. [26]. Their case study shows that the DE algorithm can generate feasible solutions with lower computational effort. The dynamic scheduling of a combined heat–power islanded microgrid with RES and energy storage was presented in Ref. [27], and a fuzzy attainment module was integrated into the modified differential evolution algorithm with dynamic mutation rates to obtain the best solution for the scheduling problem. However, most past works simplified the constraints of power balance and neglected the power loss and voltage constraints. In fact, the operation of a microgrid must be carefully controlled in order to prevent abrupt voltage fluctuations that stem from the sensitivity of voltages to variations in power injections, node over-voltages and under-voltages, and drops in the power factor [28]. As explained in Refs. [29,30], voltage stability plays an important role in power system operations. Therefore, power flow limits are advocated in this context to ensure the stability of voltage. Even though the importance of power flow constraints was emphasized in Refs. [31,32], they focused only on a specific time interval and ignored the coordination among different periods. But it is necessary to consider it over multiple periods because the states of charge (SOC) of batteries are relevant to their last SOC, and power flow distributions are period-coupling. Although the power flow constraints and the detailed network structure were considered in Ref. [6], the population size of the employed evolutionary algorithm is too large to conceal their influences on the increasing computational time.

In fact, solution techniques to optimal power dispatch problems for microgrids have always attracted scholars’ attention, particularly in recent years. On the one hand, the consideration of coordinating the power dispatch among multiple periods, as well as the complication of a transmission network, makes the optimal power dispatch problem more complex. On the other hand, the rolling horizon scheduling proposed by the research in Refs. [33,34], which was proposed to handle the uncertain nature of the power sources and load, put forward higher requirements for solving the scheduling model. The effectiveness of small population sizes in decreasing computational time has been verified in Refs. [35,36,37,38], but it may obtain bad solutions for intricate optimization problems due to its easily damaged population diversity. Hence, reducing the computational time needed to solve such an optimal power dispatch problem is still very challenging. Decreasing the population size and implementing the parallelization of the evolutionary algorithm may be an alternative way to balance this contradiction. In the parallelization of the evolutionary algorithm, the large main population, which is employed to enhance the population diversity, can be divided into several populations, each with a smaller size, and then each smaller population can be used to search for the optimal results independently and concurrently. Both the computational time, which depends on the small population size, and the population diversity, which depends on the total population, will be improved. In these circumstances, the contradiction between computational time and the precision of solutions will be resolved. Of course, it will occupy more memory space and CPU cores. Fortunately, hardware costs have become lower with the development of manufacturing technology for CPUs and ROM/RAM.

It should be noted here that many significant research efforts have been dedicated to exploring parallel implementations of differential algorithms. These efforts encompass three principal methodologies: CPU-driven parallelism [39], GPU-accelerated parallel implementations [40], and hybrid parallelization frameworks [41]. GPU implementations achieve paradigm-shifting speedups in evolutionary computation for data-parallel, large-population scenarios. However, their efficacy is contingent upon co-designing algorithms with hardware constraints, mitigating memory access contention, and PCIe transfer overheads. In contrast, CPU or heterogeneous CPU-GPU frameworks maintain advantages for fine-grained, decision-intensive problems, where branch divergence degrades GPU throughput. In this study, we concentrate on parallel computing approaches utilizing CPU architectures. The main contribution of this work is outlined as follows.

(1) An improved parallel differential evolution (PDE) approach is proposed for optimal power dispatch in microgrids, accounting for power flow limits. The proposed PDE approach is developed based on the message-passing interface (MPI) standard [42] in order to reduce the computational time and obtain a good solution.

(2) The PDE approach splits the main population into several smaller populations, and each small population performs the mutation operation and crossover operation in different CPU processes independently at the same time. The computational time of the DE algorithm using a small population size is less than that of a large population. As a consequence, the PDE algorithm will spend less computational time compared to a traditional serial DE algorithm.

(3) The gather and scatter operations are also employed to ensure the preferred population diversity. In the gather operation, the small population of each process is gathered to the root process after some given iterations, and these gathered individuals will perform the operations of the mutation, crossover, and selection to enhance the population diversity. Then, the gathered population will be scattered into several small populations again to perform a mutation and crossover in different processes.

(4) Some other improvements on the mutation, adaptive parameters, and the introduction of a migration operation are also implemented in order to find the optimal solution quickly and generate superior vectors in a more effective way. The feasibility and effectiveness of the proposed PDE approach are verified and evaluated through two test microgrid systems.

The remaining part of this paper is structured as follows. Section 2 introduces the problem formulation for the optimal power dispatch problem in a microgrid. Section 3 briefly reviews the traditional differential evolution algorithm. The proposed PDE algorithm based on an MPI standard is proposed in Section 4. Section 5 introduces the improvements of the PDE algorithm for optimal power dispatch in a microgrid. Section 6 provides numerical results, and Section 7 concludes.

2. Problem Formulation

Optimizing the objective function is very important in the optimal dispatch of a microgrid, and various prioritizing goals depending on different operational contexts have been involved. For example, Hu et al. [43] minimized the total operation cost of the involved microgrids. Sati et al. [44] optimized the total generation cost of the considered system, Ashfaq et al. [45] minimized the total power losses and voltage profile, Zhang et al. [6] optimized the total generation and operation cost of the involved microgrids, and Basu [46] aimed at optimizing the total profit function of involved microgrids. This study focuses on addressing the economic–environmental tradeoff. The corresponding objective function is formulated in detail as follows.

2.1. Objective Functions

The objective function C considered in the optimization problem is described as (1).

$$\min C=C_O+C_F+C_{EV}+C_{grid}$$

(1)

where $C$ is total cost, $C_O$ is the maintenance cost, $C_F$ is fuel cost, $C_{EV}$ is emissions treatment cost and $C_{grid}$ is the cost of exchanging power with the main grid. The maintenance cost $C_O$ is formulated as (2).

$$C_O={\displaystyle \sum _{t=1}^{24}{\displaystyle \sum _{i=1}^N{C}_{Oi}{P}_i(t)}}$$

(2)

where N is the number of micro-sources including wind turbines, photovoltaic systems, batteries, and micro-turbines, $C_{Oi}$ is the maintenance coefficient of micro-source i, and $P_i(t)$ is the active power of micro-source i at time interval t. Only the micro-turbine will produce fuel costs and emission disposal costs among those distributed generators, and its fuel cost $C_F$ can be formulated as (3).

$$C_F={\displaystyle \sum _{t=1}^{24}{\displaystyle \sum _{i=1}^L{C}_{Fi}{P}_{Fi}(t)}}$$

(3)

where L is the number of micro-turbines, $C_{Fi}$ is the coefficient of fuel cost of micro-turbine i, and $P_{Fi}(t)$ is the active power of micro-turbine i at time interval t. Micro-turbines will release pollutants, including carbon dioxide, sulfur dioxide, and nitrogen oxide. The emission cost $C_{EF}$ is formulated as (4).

$$C_{EV}={\displaystyle \sum _{t=1}^{24}{\displaystyle \sum _{k=1}^m{\displaystyle \sum _{i=1}^L{\beta }_k{\alpha }_k{P}_{MTi}(t)}}}$$

(4)

where m is the number of pollutant category, ${\alpha }_k$ and ${\beta }_k$ are the emission factor and unit price of the k-th pollutants, respectively, and $P_{MTi}(t)$ is the active power of micro-turbine i at time interval t. The cost of power trade with the main grid $C_{grid}$ is formulated as shown in (5).

$$C_{grid}={\displaystyle \sum _{t=1}^{24}(C_{Pgrid}(t)\times P_{Pgrid}(t)-C_{Sgrid}(t)\times P_{Sgrid}(t))}$$

(5)

where $C_{grid}$ is the cost of power trade between the considered microgrid and the corresponding main grid, $C_{Pgrid}$ and $C_{sgrid}$ are the purchasing and selling prices from the microgrid to the main grid, respectively, and $P_{sgrid}$ and $P_{Pgrid}$ are the selling and purchasing electricity from the microgrid to the main grid, respectively.

2.2. Constraints

(1)

Power balance constraint

The active and reactive power injections, $P_{i,n}$ and $Q_{i,n}$ of bus i in the considered microgrid is formulated as (6).

$$\left\{\begin{array}{c}{P}_{i,n}=e_i{\displaystyle \sum _{j=1}^n(G_{ij}{e}_j-B_{ij}{f}_j)}+f_i{\displaystyle \sum _{j=1}^n(G_{ij}{f}_j+B_{ij}{e}_j)}\\ Q_{i,n}=f_i{\displaystyle \sum _{j=1}^n(G_{ij}{e}_j-B_{ij}{f}_j)}-e_i{\displaystyle \sum _{j=1}^n(G_{ij}{f}_j+B_{ij}{e}_j)}\end{array}\right.$$

(6)

where $e_i$ and $f_i$ are the real and imaginary part of the voltage of bus i, respectively, and $G_{ij}$ and $B_{ij}$ are the real and imaginary part of mutual admittance between bus i and bus j, respectively. In particular, $P_{i,n}$ and $Q_{i,n}$ can be formulated as (7).

$$\left\{\begin{array}{l}{P}_{i,n}=P_{i,G}-P_{i,L}\\ Q_{i,n}=Q_{i,G}-Q_{i,L}\end{array}\right.$$

(7)

where $P_{i,G}$ and $Q_{i,G}$ are the total active and inactive power of distributed energy sources at bus i, respectively. $P_{i,L}$ and $Q_{i,L}$ are the total active and inactive power of loads at bus i, respectively.

(2)

Bus voltage amplitude constraint

The limits of bus voltage amplitude $V_i$ are formulated as (8).

$$V_{i,\min }\le V_i\le V_{i,\max }$$

(8)

where $V_i$ is the voltage amplitude of bus i, $V_{i,\min }$ and $V_{i,\max }$ are the minimum and maximum voltage amplitude of bus i, respectively.

(3)

Transformer ratio constraint

The limits of transformer ratio $k_i$ of the k-th transformer is formulated as (9).

$$k_{i,\min }\le k_i\le k_{i,\max }$$

(9)

where $k_i$ is the ratio of transformer i, $k_{i,\min }$ and $k_{i,\max }$ stands for the minimum and maximum value of the ratio for the i-th transformer, respectively.

(4)

Active power constraint

The constraints of active power generation $P_i(t)$ of the i-th generator at the t-th time interval are listed as (10).

$$P_i^{\min }\le P_i(t)\le P_i^{\max },\forall i=1,2,\dots ,H$$

(10)

where $P_i^{\min }$ and $P_i^{\max }$ are the minimum and maximum of the active power of distributed generator i, respectively, H is the number of distributed generators.

(5)

Batteries constraint

The state of charge of the i-th battery at the t-th time interval $SO{C}_i(t)$ is limited as (11).

$$SO{C}_{i,\min }\le SO{C}_i(t)\le SO{C}_{i,\max }$$

(11)

where $SO{C}_i(t)$ is the SOC of battery i at time interval t, $SO{C}_{i,\min }$ and $SO{C}_{i.\max }$ are the minimum and maximum of the SOC of battery i, respectively.

(6)

Transmission constraint

Transmission power $S_i(t)$ of branch i at time interval t is limited as shown in (12).

$$\left|S_i(t)\right|\le S_i^{\max }$$

(12)

where $S_i(t)$ is the transmission power flow of transmission line l at time interval t, $S_i^{\max }$ denotes its maximum value.

3. Brief Overview of Differential Evolution Algorithm

The differential evolution (DE) algorithm, a heuristic approach for minimizing potentially nonlinear and non-differentiable continuous space functions, was proposed by Rainer Storn and Kenneth Price in Ref. [23]. It comprises the operations of initialization, evaluation, mutation, crossover, and selection, offering advantages such as requiring few control variables and exhibiting robustness. Consider a population size of N and a variable dimensionality of D; the i-th individual at the t-th generation is denoted by Equation (13). The subsequent section details the steps of the traditional DE algorithm, several of which are incorporated into the PDE algorithm.

$$X_{i,t}=(x_{i,t}^1,x_{i,t}^2,\cdots ,x_{i,t}^D),i=1,2,\cdots ,N$$

(13)

Step 1: Initialization: the initial population of the algorithm is randomly generated in the range of $X_{\min }$ to $X_{\max }$ and the initial population is expressed as (14). The boundaries of these variables are expressed as Equations (15) and (16).

$$X_{i,0}=(x_{i,0}^1,x_{i,0}^2,\cdots x_{i,0}^D),i=1,2,\cdots ,N$$

(14)

$$X_{\min }=(x_{\min }^1,x_{\min }^2,\cdots ,x_{\min }^D)$$

(15)

$$X_{\max }=(x_{\max }^1,x_{\max }^2,\cdots ,x_{\max }^D)$$

(16)

Step 2: Calculate the fitness function value $f(X_{i,t})$ of each target vector $X_{i,t}$ in the current population.

Step 3: Mutation: For each target individual $X_{i,t}$ i = 1, 2, …, N, three mutually different integers (i.e., $r1,r2,r3\in \left\{1,2,\dots ,N\right\}$ are first generated randomly. It should be noted here that four numbers (i.e., $r1,r2,r3,i$) are different from each other. Then, mutant vector $V_{i,t}$ for the i-th target is generated according to Equation (17).

$$V_{i,t}=X_{r1,t}+F(X_{r2,t}-X_{r3,t})$$

(17)

where F is mutagenic factor and it is generated randomly between [0, 2]. If $V_{i,t}$ is not in the limited range, set $V_{i,t}=X_{\min }+rand(0,1)\ast (X_{\max }-X_{\min })$, where rand(0,1) is a random number $\in \left[0,1\right]$.

Step 4: Crossover: Generate trial vectors for all target vectors, and the j-th dimension value of a trial vector is formed by mutant vector $V_{i,t}$ and target vector $X_{i,t}$ according to Equation (18). Here, an example of the crossover operation for 5-dimensional vectors is given in Figure 1 to help understand the crossover mechanism.

$$u_{i,t}^j=\left\{\begin{array}{l}{v}_{i,t}^j ran{d}_j\le CR \mathrm{or} j=rand{n}_i\\ x_{i,t}^j ran{d}_j>CR \mathrm{and} j\ne rand{n}_i\end{array}\right.j=1,2,\dots ,D$$

(18)

where $ran{d}_j$ is a random number $\in \left[0,1\right]$, $rand{n}_i$ is a randomly chosen index $\in \left\{1,2,\dots .D\right\}$, and $CR$ is the crossover constant $\in [0,1]$.

Step 5: Selection: The i-th vector of next generation is chosen from the trial vector $u_{i,t}^j$ and the original target vector $X_{i,t}$ using the greedy criterion according to Equation (19). Within the selection operation described by (19), if the fitness value of trial vector $U_{i,t}$ is smaller than that of $X_{i,t}$, $U_{i,t}$ will be one vector of the next generation. Otherwise, $X_{i,t}$ will still be a member of next generation.

$$X_{i,t+1}=\left\{\begin{array}{l}{U}_{i,t}f(U_{i,t})<f(X_{i,t})\\ X_{i,t}f(U_{i,t})\ge f(X_{i,t})\end{array}\right.i=1,2,\cdots ,N$$

(19)

Step 6: End condition: if the given iteration number is reached, the optimal solution is outputted, and the algorithm will end. Otherwise, go to Step 2.

4. Parallel Differential Evolution Algorithm

The computational time of the traditional differential evolution (DE) algorithm is primarily governed by two factors: the dimensionality of the solution vector and the population size employed. Generally, a smaller population size reduces computational time for the same optimization problem. However, arbitrarily decreasing population size to accelerate computation is often inadvisable, as it may compromise population diversity and lead to suboptimal solutions. While population reduction remains theoretically feasible if diversity preservation is ensured, this work proposes a parallel DE algorithm based on the message-passing interface (MPI) [42] standard to concurrently address computational efficiency and diversity maintenance. The framework of the proposed parallel DE (PDE) algorithm is illustrated in Figure 2.

The PDE algorithm partitions the main population into multiple subpopulations that perform mutation, crossover, and selection operations concurrently across distinct CPU processes. This architecture ensures the algorithm’s computational time primarily depends on subpopulation size, thereby significantly reducing total execution time. After predetermined iterations, all processes synchronize information to the root process to enhance population diversity. The aggregated population undergoes mutation and crossover in the root process before being redistributed to individual processes for independent operations. This periodic gathering mechanism safeguards solution quality through diversity preservation. Here, taking a 5-dimensional vector space optimization as an example, Figure 3 demonstrates the PDE workflow distributed across three processes.

5. Improvements on Proposed PDE Approach

5.1. Sorting Vectors for Mutation

The three target vectors in the mutation operation of the traditional DE algorithm are chosen randomly. However, this is not conducive to generating good solutions. Here, a sorting strategy for the selected target vectors [39] is employed to help the algorithm search along the direction towards to the best individual selected. Within the sorting strategy, the best one among the three vectors will be selected as $r_1$, the second one is considered as $r_2$ and the last one as $r_3$.

5.2. Adaptive Parameter Updating

The scaling factor F and crossover probability factor CR are two important parameters in traditional DE, which play an important role in the search and convergence of DE. A proper choice of F and CR will help to accelerate the solving process of optimization problems. However, F and CR are usually set as constant values that have to be determined by users in the traditional DE algorithm. Generally, dynamically or self-adaptively generating F and CR will help to generate more superior vectors effectively [47]. Aiming at the issue, a self-adaptive selection for F and CR is employed in the improved PDE algorithm for the optimal power dispatch problem in a microgrid. Within the adaptive selection strategy, each individual of the population has its own F and CR, and they can be dynamically adjusted with the evolution of the population according to some specific rules. The initial F and CR in the proposed PDE are generated randomly according to their minimum and maximum values, which are given by the prior experience. Generally, the minimum and maximum values of F are 0.1 and 0.9, while they are 0.0 and 0.9 for CR. The values of F and CR for different individuals at various generations will be updated using (20) when the corresponding individual has not generated a better solution in some iterations.

$$\begin{array}{l}{F}_i^{\prime }=F_i+(1-Z_i/Z_b)\ast (F_b^k-F_i)\\ C{R}_i^{\prime }=C{R}_i+(1-Z_i/Z_b)\ast (C{R}_b^k-C{R}_i)\end{array}$$

(20)

where $F_i$ and $C{R}_i$ are the current parameters that individual i adopts at the k-th generation, $F_i^{\prime }$ and $C{R}_i^{\prime }$ are their updated parameters, $Z_i$ is the total times of individual i generating a better solution, while $Z_b$ is the maximum value among them, and $F_b^k$ and $C{R}_b^k$ are the current parameters of the corresponding $Z_b$.

5.3. Migration Operation

Generally, a fast population converging will accelerate the searching process. However, too-fast convergence will result in reducing search efficiency and improving probability of falling into a local optimum for solutions. Aiming to balance the converging velocity and hence maintain the population diversity, a migration operation proposed in Ref. [35] is adopted to alleviate the prematurity. If the crowding level, which is formulated by (21), is more than its tolerated level determined by ${\epsilon }_1$, a new diverse population is regenerated around the present optimal solution using (23) within the introduced migration operation.

$$\rho ={\displaystyle \frac{{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{d=1}^D\eta \left(X_{di}^k\right)}}}{N\ast D}}$$

(21)

where $\eta \left(X_{di}^k\right)$ denotes the closeness degree of the i-th individual to the best individual along with the d-th dimensionality in the evolution population, and it is calculated using Equation (22). N is the population size, D is the dimensionality of individual, $X_{di}^k$ is the d-th dimension parameter of individual i at the k-th generation.

$$\eta \left(X_{di}^k\right)=\left\{\begin{array}{l}1\hspace{1em}\left|X_{di}^k-X_{db}^k\right|/\left|X_{db}^k\right|<{\epsilon }_2\\ 0\hspace{1em}others\end{array}\right.$$

(22)

where $X_{db}^k$ is the d-th dimension parameter of the best individual at the k-th generation, ${\epsilon }_2$ is a constant to measure of the approaching distance between the best individual and the other individuals. Based on experience, the value of parameter ${\epsilon }_2$ should not exceed 0.3, and the value of parameter ${\epsilon }_1$ should not be lower than 0.6. Here, ${\epsilon }_2$ is set to 0.15 and ${\epsilon }_1$ is set to 0.85 according to [35]. Within the introduction of migration operation, the population diversity of the evolutionary population will be maintained because once the population comes together around the best individual, the population will be initially generated around the best individual. It is precisely through this migration that the diversity of the population is maintained to a certain extent.

$${X_{di}^k}^{\prime }=\left\{\begin{array}{l}{X}_{db}^k+{\omega }_1\ast ({\left.X_d^k\right|}_{\max }-X_{db}^k) {\omega }_2>{\displaystyle \frac{{\left.X_d^k\right|}_{\max }-X_{db}^k}{{\left.X_d^k\right|}_{\max }-{\left.X_d^k\right|}_{\min }}}\\ X_{db}^k+{\omega }_1\ast (X_{db}^k-{\left.X_d^k\right|}_{\min }) otherwise \end{array}\right.$$

(23)

where $X_{db}^k$ is the d-th dimension value of the best individual at the k-th generation while $X_{di}^{k\prime }$ denotes its new generated value, ${\omega }_1$ and ${\omega }_2$ are the uniformly random numbers distributed in [0, 1], and ${\left.X_d^k\right|}_{\max }$ and ${\left.X_d^k\right|}_{\min }$ are the maximum and minimum boundary values of the d-th dimension parameter of the population at the kth generation.

5.4. Dispatch Strategy

The distributed generators involved in the microgrid consist of wind generators, photovoltaic systems, batteries, and micro-turbines. However, the outputs of photovoltaic systems and wind generators are uncontrollable due to the max-power trapping control modes employed. Only the outputs of batteries and micro-turbines need to be dispatched in the microgrid. The dispatch strategy for controllable generators depends on the active power of the slack bus. The active power of the slack bus donated by $P_{s,t}$ is obtained through power flow calculation. If the $P_{s,t}$ is beyond its limited range, the outputs of batteries and micro-turbines in other PV buses will be adjusted. The steps of adjustment are shown as follows.

Step 1: Calculate the $delta{P}_{s,t}$ according to (24).

$$delta{P}_{s,t}=\left\{\begin{array}{l}{P}_{s,t}-P_{s,t}^{\max }\hspace{1em}if\hspace{0.33em}{P}_{s,t}>P_{s,t}^{\max }\\ P_{s,t}-P_{s,t}^{\min }\hspace{1em}if\hspace{0.33em}{P}_{s,t}<P_{s,t}^{\min }\end{array}\right.$$

(24)

where $P_{s,t}^{\min }$ and $P_{s,t}^{\max }$ are the minimum and maximum active power of the slack bus, respectively.

Step 2: If $abs\left|delta{P}_{s,t}\right|>f_1$ where $f_1$ is a constant to determine whether the batteries are adjusted, adjust the power outputs of batteries according to (25). Otherwise, go to Step 6.

$$P_{Bi}{(t)}^{\prime }=P_{Bi}(t)+{\displaystyle \frac{{\lambda }_1\ast delta{P}_{s,t}}{{\lambda }_2\ast quaB}}$$

(25)

where $P_{Bi}(t)$ is the original active power of battery $i$ at time interval $t$, $P_{Bi}{(t)}^{\prime }$ is the adjusted active power, ${\lambda }_1$ is a factor which satisfies ${\lambda }_1>1$ in consideration of the power loss, ${\lambda }_2$ is a factor to reduce the adjustment amount every time, avoiding going beyond the ranges of SOC of the batteries, and $quaB$ is the number of batteries.

Step 3: Rectify the adjusted power of batteries to satisfy the constraints of battery outputs and then carry out the power flow calculation using the new adjusted power.

Step 4: If the new $abs\left|delta{P}_{s,t}\right|>f_2$ where $f_2$ is a constant to determine whether the micro-turbines are adjusted, adjust the power outputs of the micro-turbine according to (26). Otherwise, go to Step 6.

$$P_{\mathrm{M}\mathrm{T}i}(t{)}^{\prime }=P_{\mathrm{M}\mathrm{T}i}(t)+{\displaystyle \frac{{\lambda }_3\ast delta{P}_{s,t}}{{\lambda }_4\ast quaT}}$$

(26)

where $P_{\mathrm{M}\mathrm{T}i}(t)$ is the original active power of micro-turbine $i$ at time interval $t$, $P_{\mathrm{M}\mathrm{T}i}(t{)}^{\prime }$ is the adjusted active power, ${\lambda }_3$ is a factor which satisfies ${\lambda }_3>1$ in consideration of the power loss, ${\lambda }_4$ is a factor to reduce the adjustment amount every time, avoiding going beyond the output ranges of the micro-turbines, $quaT$ is the quantity of micro-turbines.

Step 5: Rectify the adjusted power of micro-turbines to satisfy the constraints of micro-turbine outputs, and then carry out the power flow calculation using the new adjusted power.

Step 6: Calculate the purchasing or selling power from the main grid. Based on the purchasing or selling power, calculate the $C_{grid}$ according to (5), and end the dispatch strategy.

5.5. Rules Dealing with Constraints

It is very hard to deal with the complex constraints involving equations and non-equations of the optimal power dispatch problem in a microgrid. The proposed PDE algorithm employs three rules for dealing with the constraints in order to obtain a feasible solution. The first rule is the frontier rule, which aims to deal with inequality constraints for decision variables. The second rule is the power flow rule, which deals with power balance constraints. The third rule is the penalty rule, which punishes infeasible solutions.

(1)

Frontier Rule

For the control variables of the optimization problem, i.e., the voltage amplitudes of PV bus, the active powers of all micro-sources, the states of charge of batteries, and the transformer ratios, if the calculated values are more than their maximums, they will be set equal to their maximums. If the values are less than their minimums, they will be set equal to their minimums.

(2)

Power Flow Rule

This rule employs the Newton–Raphson method to implement power flow calculation through which all state variables of the candidate solution will be calculated. Here in this work, the Newton–Raphson method expressed by rectangular coordinates is adopted. Once the convergence of the method is achieved successfully, the handling processes for the constraints will be finished. Otherwise, the penalty rule, which will be introduced in the next paragraph, will be considered. The method of power flow calculation is introduced in Ref. [35] in detail.

(3)

Penalty Rule

The penalty rule has to be employed to help individuals getting rid of infeasible regions, while the above two rules fail. In fact, the penalty rule will be used when a solution violates the power flow constraint or the bus voltage amplitude constraint. The total accumulated penalty terms of a candidate solution at the k-th generation will be calculated as (27).

$$penalt{y}_k={\displaystyle \sum _{t=0}^T\left(a{f}_{bv,k}\ast V{A}_{bv,k,t}+a{f}_{pf,k}\ast V{A}_{pf,k,t}\right)}$$

(27)

where k is the iteration number, $a{f}_{bv,k}$ is the amplification factor of violating the bus voltage amplitude constraint in the k-th iteration; $V{A}_{bv,k,t}$ is the accumulated violated amount of all bus voltage amplitude constraints at time interval t in the corresponding iteration; $a{f}_{pf,k}$ is the amplification factor of violating the power flow constraint; $V{A}_{pf,k,t}$ is the accumulated violated amount of all power flow constraints at time interval t.

The amplification factors $a{f}_{i,k}$ are dynamic and are formulated as (28). With the increase in iterations, the amplification factor is linear growth. As a consequence, the impact of the penalty function on the total cost is small in early iterations and enormous in the latter iterations.

$$a{f}_{i,k}=a{f}_{i,\min }+(a{f}_{i,\max }-a{f}_{i,\min })\ast k/k_{\max }$$

(28)

where $k_{\max }$ is the given maximal iterations; $a{f}_{i,k}$ is the amplification factor in the k-th iteration for violation of the constraint i; $a{f}_{i,\min }$ and $a{f}_{i,\max }$ are the minimum and maximum amplification factors for violation of the constraint i, respectively.

6. Case Study and Result Analysis

The effectiveness of the proposed PDE approach for optimal power dispatch problems in a microgrid is verified through two test systems whose power generators include photovoltaic systems, wind generators, batteries, and micro-turbine generators. Additionally, larger-scale test systems that are modified through the second test system but with more optimizing variables are also employed to test the proposed approach in detail.

6.1. Test Systems and Parameter Settings

The involved two test systems, one with IEEE 9-bus and the other with IEEE 39-bus networks [48], are applied to verify the effectiveness of the proposed PDE approach. The entire scheduling period is one day, which is divided into 24 intervals. The detailed configurations of the two systems are shown in

Table 1. Configurations of test system 1 with 9-bus network.

Bus	Type	Number	Bus	Type	Number
0	Battery	6	1	Wind generator 1	3
0	Micro-turbine 1	2	1	Wind generator 2	6
0	Micro-turbine 2	2	2	Photovoltaic	8
0	Micro-turbine 3	2

and

Table 2. Configurations of test system 2 with 39-bus network.

Bus	Type	Number	Bus	Type	Number
29	Photovoltaic	3	33	Wind generator 1	6
30	Batteries	3	34	Wind generator 1	9
30	Micro-turbine 1	2	35	Photovoltaic	5
30	Micro-turbine 2	2	35	Batteries	2
31	Wind generator 2	10	36	Photovoltaic	5
31	Batteries	3	36	Batteries	2
32	Wind generator 2	10	37	Micro-turbine 2	2
32	Batteries	3	38	Micro-turbine 1	2

, respectively, while the corresponding parameters of different micro-sources are shown in

Table 3. Parameters of different micro-sources.

Micro-Sources	Parameters
Wind 1	$C_{oi}=1.875$ CNY/kWh, Cut-in speed = 3 m/s, Cut-out speed = 25 m/s, Rated speed = 11 m/s, Rated power = 0.18 × 102 kW
Wind 2	$C_{oi}=1.875$ CNY/kWh, Cut-in speed = 3 m/s, Cut-out speed = 25 m/s, Rated speed = 13 m/s, Rated power = 0.18 × 102 kW
Photovoltaic	$C_{oi}=2.65$ CNY/kWh, Temperature coefficient = 0.0041, Rated power = 0.4 × 102 kW, Standard radiation = 1 kW/m2, Temperature = 25 °C
Batteries	$C_{oi}$ = 0.095 CNY/kWh, $SO{C}_{\min }$ = 0.3, $SO{C}_{\max }$ = 1
Micro-turbine 1	$C_{oi}$ = 0.608 CNY/kWh, $C_{Fi}$ = 0.211 CNY/kWh, ${\displaystyle \sum {\alpha }_k{\beta }_k}$ = 0.165 CNY/kWh, $P_{\max }$ = 3 × 102 kW, $P_{\min }$ = 0 kW
Micro-turbine 2	$C_{oi}$ = 0.589 CNY/kWh, $C_{Fi}$ = 0.211 CNY/kWh, ${\displaystyle \sum {\alpha }_k{\beta }_k}$ = 0.18 CNY/kWh, $P_{\max }$ = 2.5 × 102 kW, $P_{\min }$ = 0 kW
Micro-turbine 3	$C_{oi}$ = 0.614 CNY/kWh, $C_{Fi}$ = 0.211 CNY/kWh, ${\displaystyle \sum {\alpha }_k{\beta }_k}$ = 0.166 CNY/kWh, $P_{\max }$ = 2.8 × 102 kW, $P_{\min }$ = 0 kW

. The varied loads for the two systems are shown in Figure 4 and Figure 5, respectively. The illumination and wind speed of photovoltaic and wind generators on the considered day are shown in Figure 6. The purchasing electricity prices for the peak period (11 h–15 h and 19 h–21 h), flat period (8 h–10 h, 16 h–18 h and 22 h–23 h), and valley period (0 h–7 h) are 0.83, 0.49 and 0.17 CNY/kWh, while the selling prices are 0.65, 0.38 and 0.13 CNY/kWh, respectively. The proposed approach, as well as other comparative algorithms, are all implemented using the C++ programming language in Ubuntu 22.04, with an Xeon^® E5-2670 v2 2.5 GHz CPU and RAM of 128 GB. The parameters, including the original ones of the baseline algorithm of the traditional DE and the additional parameters, which consist of the small population size (SPS) as well as the parallel process number (PPN), and the gather and scatter frequency (GSF) of the proposed PDE algorithm for the two test systems, are shown in

Table 4. Parameter settings of PDE algorithm for different systems.

System	Parameters
Test System 1, 2	$F^{\min }=0.1, F^{\max }=0.9, C{R}^{\min }=0.0, C{R}^{\max }=0.9, {\lambda }_1=1.3, {\lambda }_2=10, {\lambda }_3=2, {\lambda }_4=1$ $f_1=3, f_2=0.5, {\epsilon }_1=0.85, {\epsilon }_2=0.15, SPS=10, PPN=10, GSF=10$
Test System 1	$a{f}_{1,\min }=\mathrm{15,000}, a{f}_{1,\max }=\mathrm{20,000}, a{f}_{2,\min }=1500, a{f}_{2,\max }=2000$, $k_{\max }=800$
Test System 2	$a{f}_{1,\min }=\mathrm{10,000}, a{f}_{1,\max }=\mathrm{15,000}, a{f}_{2,\min }=200, a{f}_{2,\max }=500$, $k_{\max }=1500$

. It should be noted here that all parameters of the proposed PDE here are not the fine-tuned ones, which means they are not the best ones but are just set through trial and error.

6.2. Results and Comparative Analysis

6.2.1. Optimization Result Presentation

Within the above parameter settings shown in Table 4, we run the proposed PDE algorithm 10 times independently for each test system, and the obtained statistical metrics, such as the best, worst, and average objective functions, as well as their standard deviation values (denoted as STDV) and the average CPU time, are shown in

Table 5. Obtained metrics of the proposed PDE for test systems I and II.

Metrics	Best	Worst	Average	STDV	AVG CPU Time
Test System I	13,953.40	14,120.10	14,035.94	60.47	0:00:20
Test System II	22,657.30	23,129.20	22,871.98	206.08	0:45:52

. It is clearly seen from the table that the best values of the obtained solutions are CNY 13,953.40 and 22,657.30 for the considered test systems, respectively. The average values of the obtained optimal cost in 10 independent runs are CNY 14,035.94 and CNY 22,871.98 for test systems I and II. The average CPU time consumption for these two systems is 27 s and 45 min 52 s for test systems I and II. The optimal SOC of different batteries and active power outputs of different micro-turbine generators for test system one are illustrated in Figure 7 and Figure 8, respectively. The obtained optimal voltage magnitudes and phase degrees of different buses of test system I are shown in Figure 9 and Figure 10, while other state variables can be easily calculated within these given decision variables through power flow calculations. It is clearly seen from these figures that all obtained control variables for the optimal solution are within their boundary limits. Accordingly, the optimal SOCs of different batteries and active power outputs of various micro-turbine generators for test system II are shown in Figure 11 and Figure 12, respectively. The obtained optimal voltage magnitudes and phase degrees of various generator buses are shown in Figure 13 and Figure 14. It is found from Figure 11 and Figure 12 that the SOCs of all batteries lie in [0.3, 1.0], and the active power outputs of all micro-turbine generators also satisfy their constraints. Additionally, the varying violations in bus voltage constraints, power flow constraints, and the corresponding penalty terms for test systems one and two are shown in Figure 15 and Figure 16 and deeply show the effectiveness of the proposed PDE algorithm. It is clearly found from Figure 15 that most violations are zero after 200 iterations for test system I, and from Figure 16, it can be seen that the violations are close to zero after 350 iterations for test system II. It is also found from these two figures that the final penalty terms converge to a relatively small value, which is infinitely close to zero after 200 and 350 iterations. That means the proposed PDE approach can effectively obtain feasible solutions. This also shows the validity of the constraint handling rules.

6.2.2. Comparisons with Baseline Algorithm of Traditional DE

To further highlight the performance of our proposed algorithm, we implemented its baseline algorithm—the traditional serial differential evolution for solving microgrid optimal scheduling problems on the same platform. Both the traditional DE and the improved PDE employ the same parameters, and they are independently run 10 times, and their comparative results are presented in

Table 6. Statistical metrics obtained by the improved PDE and the baseline algorithm of traditional DE for test system I.

Metrics	Best	Worst	Average	STDV	AVG CPU Time
Improved PDE	13,953.4	14,120.1	14,035.9	60.47	0:00:22
Traditional DE	13,966.6	14,350.7	14,099.6	154.17	0:02:54

and

Table 7. Statistical metrics obtained by the improved PDE and the baseline algorithm of traditional DE for test system II.

Metrics	Best	Worst	Average	STDV	AVG CPU Time
Improved PDE	22,657.3	23,129.2	22,872.0	206.08	0:45:52
Traditional DE	23,198.2	35,632.5	30,503.3	5047.12	5:26:57

for the considered test systems, respectively. Additionally, the convergence characteristics for the obtained optimal solutions of the two algorithms are shown in Figure 17 and Figure 18 for test systems I and II, respectively.

It is shown from Table 6 that the best, worst, average, and standard deviation values of the improved PDE are CNY 13,953.4, 14,120.1, 14,035.9, and 60.47, which are a little smaller than what the traditional DE obtained at the corresponding values of CNY 13,966.6, 14,350.7, 14,099.6, and 154.17. Additionally, it is quite obvious from the table that the average CPU time at the value of 22 s consumed by the proposed PDE is roughly smaller than that of the traditional DE at the value of 2 min 54 s. This shows the effects of the proposed IDE algorithm. According to the results shown in Table 7, the improved PDE obtains much better results than the traditional DE Additionally, and obviously, the consumed CPU time of the proposed improved PDE is relatively smaller than that of its baseline algorithm—the DE algorithm—in which the proposed PDE only consumes 45 min 52 s, while the traditional DE consumes about 5 h 26 min and 57 s. Additionally, the multi-core hardware (such as the number of CPU cores and memory occupation) for the proposed PDE and the traditional DE are also reported in

Table 8. Multi-core hardware occupation of the involved algorithms.

Systems	Methods	CPU Cores	Initial Resident Memory	MEM (%)	CPU (%)
Test System I	TDE	1	26,364 K	0.0206	100
	PDE	10	28,408 K	0.0222	100
Test System II	TDE	1	26,384 K	0.0206	100
	PDE	10	27,284 K	0.0214	100

. It is clearly seen from this table that each process of these two algorithms occupies approximately equivalent memory space, but the total resources of all CPU cores for the proposed PDE are about 10 times those of the traditional DE. The implementation of this proposed parallel approach adheres to the space–time tradeoff principle, strategically consuming additional memory and CPU resources to achieve significant computational speedups. Fortunately, the price of disks and CPUs is becoming cheaper with the development of technology at present.

It is shown from Figure 17 that the PDE outperforms the traditional DE after 200 iterations for test system I. Moreover, as shown in Table 6, the CPU time consumption of the PDE is significantly lower than that of the traditional DE. In one world, though these two algorithms obtain nearly close results, the traditional DE consumes about 175 s CPU time, while the PDE algorithm only consumes 22 s. In contrast, for test system II, it is quite clearly seen from Figure 18 that the proposed PDE approach yields the best convergence nature and obtains better fitness than the baseline algorithm does. Hence, it can be concluded that the PDE algorithm outperforms the traditional DE, not only in terms of convergence but also in terms of convergence cost.

Additionally, a Wilcoxon rank sum test is implemented for the considered two test systems in order to compare the statistical performances of these two algorithms. Here, the null hypothesis is that the solutions obtained by the improved PDE and the traditional serial DE are samples from continuous distributions with equal medians. The test is carried out at a significance level of 0.05 between the proposed PDE and traditional DE, and the test results are shown in

Table 9. Wilcoxon rank sum test of improved PDE and traditional DE for test systems I and II.

System	p	h	Results
Test System I	0.3832	0	PDE obtains similar solution distribution with traditional DE
Test System II	6.66 × 10−4	1	PDE obtains different solution distribution with traditional DE

. In this table, p returns a significant probability, representing whether the improved PDE has the same distribution as the corresponding traditional DE, and h returns the result of the hypothesis test. If the overall difference between the improved PDE and the traditional DE is not significant, then h is zero; if the overall difference between the improved PDE and the traditional DE is significant, then h is one. If p is close to zero, then the null hypothesis can be questioned, which means the improved PDE generates a different solution distribution from the compared algorithm. It is quite clearly seen in Table 9 that p = 0.3832 and h = 0 for test system I, and this means the PDE obtains a similar solution distribution as the traditional DE does for this test system. As for test system II, it is quite obvious from this table that the p value is 6.66E-4 and h is equal to one, which means the proposed PDE obtains a different solution distribution from the traditional DE.

6.3. Parameter Sensitivity Analysis

There are three additional parameters in the proposed PDE algorithm, including the small population size (SPS), the parallel processes number (PPN), and the gather and scatter frequency (GSF). In order to show the impacts of different values of SPS, PPN, and GSF on solution effectiveness, the following three cases are designed.

Case 1: various values of SPS when PPN is 10 and GSF is 10.

Case 2: various values of PPN when SPS is 10 and GSF is 10.

Case 3: various values of GSF when SPS is 10 and PPN is 10.

Here, taking test system II as a sample, each scheme runs 10 times, and the obtained statistical metrics in 10 independent runs are reported in

Table 10. Results using different SPS for case 1 of test system 2.

SPS	Best (CNY)	Worst (CNY)	Mean (CNY)	CPU Time (min)
5	23,121.5	23,507.6	23,288.6	21.3
10	22,657.3	23,129.2	22,872.0	45.7
15	22,577.8	23,055.6	22,838.9	69.0

,

Table 11. Results using different PPN for case 2 of test system 2.

PPN	Best (CNY)	Worst (CNY)	Mean (CNY)	CPU Time (min)
5	22,997.8	23,508.0	23,296.1	40.3
10	22,657.3	23,129.2	22,872.0	45.7
20	22,480.0	22,735.9	22,572.1	72.2

and

Table 12. Results using different GSF for case 3 of test system 2.

GSF	Best (CNY)	Worst (CNY)	Mean (CNY)	CPU Time (min)
5	23,054	23,261.2	23,191.3	43.14
10	22,657.3	23,129.2	22,872.0	45.68
15	22,607.3	23,439.4	23,008.1	43.46
25	23,677.4	25,450.6	24,577.5	46.14
50	24,835.9	27,473.4	26,461.3	43.26
100	26,846.6	28,368.0	27,688.3	47.25

. It can be found from Table 10 that the larger the SPS the PDE employs, the better the solution that will be obtained. However, a larger SPS takes more CPU time. Regarding the parameter of PPN, it is shown in Table 11 that with the increase in PPN, the solution is better. However, it also takes more CPU time.

As for the impacts of the parameter GSF reported in Table 12, we can conclude the following. On one hand, there exists no obvious linear relationship between the GSF and the CPU time consumed. This means that the CPU time consumed by the operation of gather and scatter is a small portion of the total CPU time. On the other hand, with the increase in GSF from 15, the solution is becoming worse because the population diversity may be worse with the increasing GSF value. In other words, the gather and scatter operation performs well in terms of balancing the population diversity.

6.4. Test for Microgrid Dispatching Problems with More Large-Scale Optimizing Variables

A modified configuration test system is employed to test the effects of the proposed approach in larger-scale optimization problems of microgrids. This larger-scale system has the same network as test system II but more controlled micro-sources. The detailed configuration of this system is shown in

Table 13. Configurations of larger-scale test system with 39-bus network.

Bus	Type	Number	Bus	Type	Number
29	Photovoltaic	3	33	Wind generator1	3
30	Batteries	6	34	Wind generator1	3
30	Micro-turbine1	4	35	Photovoltaic	3
30	Micro-turbine2	4	35	Batteries	4
30	Micro-turbine3	2	36	Photovoltaic	3
31	Wind generator2	5	36	Batteries	4
31	Batteries	6	37	Micro-turbine2	4
32	Wind generator2	5	38	Micro-turbine1	4
32	Batteries	6

, and within this configuration, the number of optimization variables is equal to 1584, which is the sum number of batteries, micro-turbine generators, transformers and generator buses multiplied by the number of time intervals. The load of various buses is multiplied by 1.5 times the corresponding original ones in test system II. In addition to the improved PDE and traditional DE, another famous algorithm, particle swarm optimizer (PSO), has also been implemented for this test system. The comparative results of these three algorithms for this test system are shown in

Table 14. Comparative results of the proposed PDE with traditional DE and PSO.

Methods	Best Solution		Worst Solution		Avg CPU Time
	Obj (CNY)	Voltage Violations (p.u.)	Obj (CNY)	Voltage Violations (p.u.)
PDE	27,104.00	0	37,769.80	0	0:41:09
TDE	26,637.31	0.01055	40,089.26	0.02399	6:28:31
PSO	41,958.18	0.00114	45,692.04	0	6:02:22

. It is nearly certain from the table that the proposed PDE can obtain feasible optimal solutions, but the other two algorithms cannot. Moreover, the average CPU time consumption obtained by the proposed PDE approach is much smaller than those values obtained by the traditional DE and PSO.

7. Conclusions

In this work, an optimal power dispatch problem in a microgrid is formulated with consideration of the transmission network and power flow constraints. An improved parallel different evolution approach is proposed in order to solve this complicated constrained optimization problem. The proposed PDE approach divides the main population into several small populations in order to save computational time. Each small population performs mutations, crossovers, and selections in different CPU processes at the same time. These different CPU processes deliver data and information effectively to the root process after some given iterations. Then, the information of all populations is collected into the root process to select a better next-genus group to ensure population diversity. Although the PDE algorithm saves on computational time, mass memory is occupied, and multi-core CPUs are used. Fortunately, with the development of science and technology, especially the development of the manufacturing technology of CPUs and ROM/RAM, hardware costs are becoming lower. It is economically feasible to reduce the computational time by using multi-core CPUs and occupying memory. The efficiency of the proposed PDE is verified by using two test systems with IEEE 9-bus and 39-bus network models, respectively. The traditional serial DE algorithm is also implemented for the same systems. The simulations and comparisons show that the proposed PDE algorithm not only saves computational time but also obtains better solutions.