Solving Optimal Power Flow Problem via Improved Constrained Adaptive Differential Evolution

Abstract

The optimal power flow problem is one of the most widely used problems in power system optimizations, which are multi-modal, non-linear, and constrained optimization problems. Effective constrained optimization methods can be considered in tackling the optimal power flow problems. In this paper, an $ϵ$-constrained method-based adaptive differential evolution is proposed to solve the optimal power flow problems. The $ϵ$-constrained method is improved to tackle the constraints, and a p-best selection method based on the constraint violation is implemented in the adaptive differential evolution. The single and multi-objective optimal power flow problems on the IEEE 30-bus test system are used to verify the effectiveness of the proposed and improved $\epsilon$adaptive differential evolution algorithm. The comparison between state-of-the-art algorithms illustrate the effectiveness of the proposed and improved $\epsilon$adaptive differential evolution algorithm. The proposed algorithm demonstrates improvements in nine out of ten cases.

1. Introduction

The optimal power flow (OPF) is a key problem in power system operation and control, which aims to minimize the objective function by optimizing the control variables in energy transportation and production. In the OPF, the operation cost is a fundamental basic goal, with an emphasis on sustainability and safety, the emissions and real power loss, as well as voltage stability, are also taken into account. In addition to the generator of active power, voltage magnitude, transformer tap and shunt capacitors, which can be directly controlled and need to be varied within a certain range, it is also necessary to ensure that the line loading and voltage of buses satisfy the safety constraints. Meanwhile, the presence of the equality constraints makes the OPF problem become a highly complex optimization problem. Hence, it is necessary to develop an effective algorithm to deal with OPF problems. The key objective of the OPF problem is to deal with the constraints, where the objective function should be optimized at the same time. Evolutionary algorithms have been employed to tackle the OPF variants [1,2,3,4]. Ida et al. [2] applies the multi-objective horse herd optimization to OPF. In the paper, a mechanism is proposed to update the constraint matrix in handling constraints. Bouchekara [3] improves electromagnetic field optimization (EFO) algorithm by using chaotic numbers instead of random numbers in calculation of the force of electromagnetic particles. Mugemanyi et al. [4] applied the bat algorithm in OPF. The proposed algorithm introduced the chaotic sequences to enhance its global search ability. Saha et al. [5] proposed the symbiotic organism search-based hybrid differential evolution and applied to OPF. Farhat et al. [6] proposed the neighborhood dimension learning search strategy-based slime mould algorithm and applied to OPF. Abdel-satter et al. [7] applied the improved salp swarm algorithm (ISSA) to OPF. Taher et al. applied an improved moth flame optimization to OPF. Akdag [8] improved the Archimedes optimization algorithm and applied it to OPF. Abbasi et al. [9] combined the harmony search (HS) algorithm with the differential evolution algorithm. This method divides decision variables into ordered spans and searches. Teeparthi and Binod Kumar [10] improved the artificial physics optimization (APO) by fuzzifying the gravitational constant (G). EI-Fergany and Hasanien [11] improved the slap swarm algorithm (SSA) to tackle OPF problems. Naderi et al. [12] integrated the particle swarm optimization (PSO) with a differential evolution, and applied it to multi-objective OPF problems. Fuzzy rules help to dynamically set the inertia factor in iterations. Four fitness functions and a voltage stability based on the model analysis are considered. Elatter et al. [13] applied the Manta Ray Foraging Optimizer in OPF, where a method of selecting an ideal solution is used in the proposed algorithm. Kahraman et al. [14] introduced the crowding distance into the manta ray foraging optimization.The proposed algorithm shows a competitive performance in complex multi-objective problems. In recent years, many improved algorithms have been proposed and applied to OPF. Gong et al. [1] proposed an adaptive differential evolution to deal with the OPF, which is quite effective. Attia [15] combined the Levy flight with the sine–cosine algorithm to enhance the global searching ability, aiming at improving the efficiency of power systems. Flexible AC Transmission Systems (FACTs) are proposed, which determines the optimal location and size of FACTS devices [16]. Taking the optimal location and size of some devices into consideration, Mohamed et al. [17] combined the moth–flame optimization algorithm and gradient-based optimizer to solve OPF. This paper presents a performance evaluation of the $ϵ$ method of the constraint handling in OPF problems. The main contribution of this paper is as follows. A constrained adaptive differential evolution is employed to address the OPF problems. A constrained mutation operator is embedded in the algorithm, which enables us to consider the constraint violation and the objective at the same time. Then, the proposed algorithm is used to tackle the OPF problems, and the experimental results illustrate the effectiveness of the algorithm.

The rest of this paper is organized as follows. Section 2 presents the proposed constrained adaptive differential evolution. Section 3 gives the mathematical model of OPF. The experimental results are presented in Section 3.1.3. Section 4 concludes the paper.

2. Adaptive Differential Evolution Based on Improved pbest Selection

Similar to genetic algorithms, the DE consists of four operations: initialization, mutation, crossover and selection. The adaptive differential evolution (JADE) is an effective DE variant. Compared with the classical DE, both the mutation and crossover parameters in JADE are changed adaptively. A new selection strategy for the best individual is proposed to increase the population diversity. The framework of the classical JADE can be presented as follows.

2.1. Initialization

In DE, a target vector $x_{i,G}$ consisting of D dimensions is defined as an individual, where G denotes the current iteration number. $x_{i,G}$ represents a decision variable vector of the problem. During the initialization phase, $NP$ individuals are randomly generated within the search range. Therefore, an individual represents a solution set of the constrained optimization problem. The formulation of initialization is as follows:

$$\begin{array}{c}\hfill x_{i,1}^j=x_{min}^j+rand(0,1)·(x_{max}^j-x_{min}^j)\end{array}$$

(1)

where $x_{i,1}$ represents the vector of the first generation of the i-th individual. $x_{min}^j$ and $x_{max}^j$ denotes to the lower and upper bound of the j-th dimension, respectively. $i=1,...,NP$, $j=1,...,D$. $rand(0,1)$ represents a random number between 0 and 1.

2.2. Mutation

DE has different mutation operations to choose from. “DE/rand/1” is the most commonly used mutation strategy, which can be formulated as:

$$\begin{array}{c}\hfill v_{i,G+1}=x_{r1,G}+F·(x_{r2,G}-x_{r3,G}),r1\ne r2\ne r3\ne i\end{array}$$

(2)

where $v_{i,G+1}$ denotes the mutation vector. $x_{r1,G}$, $x_{r2,G}$ and $x_{r3,G}$ are three individuals randomly chosen from the G-th generation of the population. F is a mutation control parameter.

In JADE, a new mutation strategy called ”DE/current-to-pbest/1” has been proven to take into account the convergence speed and global search ability, which can be formulated as:

$$\begin{array}{c}\hfill v_{i,G+1}=x_{i,G}+F·(x_{best,G}-x_{i,G})+F·(x_{r1,G}-x_{r2,G}),r1\ne r2\ne i\end{array}$$

(3)

where $x_{best,G}$ denotes the top 100p% individuals in the current population. F is a random number that follows the Cauchy distribution with a mean of $u_F$ and a standard deviation of 0.1. Usually $u_F$ is initialized to 0.5.

After the mutation and crossover, the collection of all successful mutation factors $F_i$ at each generation is denoted by $S_F$. Then, $u_F$ is updated as follows:

$$\begin{array}{c}\hfill u_F=(1-c)·u_F+c·mea{n}_L\left(S_F\right)\end{array}$$

(4)

where $mea{n}_L\left(S_F\right)$ indicates that the Lehmer mean of the $S_F$, c is a constant between 0 and 1.

2.3. Crossover

Crossover(CR) operations create trial vectors $u_{i,G+1}$ by randomly selecting components from the target vector $x_{i,G}$ or its mutant vector $v_{i,G+1}$. The binomial crossover is typically described as

$$\begin{array}{c}\hfill u_{i,G+1}^j=v_{i,G+1}^j,ifrand<C{R}_{i}orj=n_j\end{array}$$

(5)

where $n_j$ is an integer random from 1 to D, ensuring that at least one dimension originates from the Vi mutation vector. $C{R}_i$ is the crossover rate of the i-th individual, which follows the normal distribution y distribution with a mean of $u_{CR}$ and a standard deviation of 0.1.

Similar to $S_F$, $S_{CR}$ records the $C{R}_i$ of successful individuals in each generation. The $u_{CR}$ is updated as follows:

$$\begin{array}{c}\hfill u_F=(1-c)·u_{CR}+c·mea{n}_A\left(S_{CR}\right)\end{array}$$

(6)

where $mea{n}_A$ indicates the arithmetic mean of $S_{CR}$.

In JADE, $C{R}_i$ is randomly generated and assigned to each individual. Gong et al. [18] proposed a CR sorting mechanism, which relates the CR value to individual fitness. In this process, normally distribute CR values are first randomly generated, and then the CR values and individuals fitness are arranged in ascending order, and each CR value is then assigned to an individual in order. Therefore, individuals with better fitness will assign smaller CR values, which can help the information from better individuals to survive into the next generation.

2.4. Improved $\epsilon$ Method

In the differential evolution, greedy strategies are applied to select offspring from a target vector $x_{i,G}$ and a trial vector $u_{i,G+1}$. The one with the better fitness value will survive into the next generation. The selection formula is as follows:

$$\begin{array}{c}\hfill x_{i,G+1}=\left\{\begin{array}{cc}{u}_{i,G+1},\hfill & iff\left(u_{i,G+1}\right)\le f\left(x_{i,G+1}\right)\hfill \\ X_{i,G},\hfill & otherwise\hfill \end{array}\right.\end{array}$$

(7)

where f denotes the fitness of the individual.

The selection operator in the proposed algorithm is based on the improved $ϵ$-constrained methods. The $ϵ$-constrained methods will exaggerate the feasible region at the beginning of the evolutionary process and then narrow it down until it meets the original feasible region. In the classical $ϵ$-constrained method proposed by Taka [19], the $ϵ$-level control strategy is as follows:

$$\left\{\begin{array}{cc}\hfill ϵ\left(0\right)& =\varphi \left(x_{\theta }\right)\hfill \\ \hfill ϵ\left(t\right)& =\left\{\begin{array}{c}ϵ\left(0\right)·{(1-G/T)}^{cp},0<G<T\hfill \\ 0,G\ge T\hfill \end{array}\right.\hfill \end{array}\right.$$

(8)

where the initial value of $ϵ$ is based on the population constraint violation level. The initial value of $ϵ\left(0\right)$ is set as the top $\theta$th individual’s constraint violation. The G-th generation $ϵ$ value is proportion to the t and initial value of $ϵ\left(0\right)$. When the generation G reaches a certain value T, the $ϵ\left(t\right)$ will be set as zero. In this way, the infeasible solutions will be guided to the feasible region along the evolution process. The setting of the $ϵ$ value is improved with the following adaptive method: However, this method ignores the information about the current population. The framework of the proposed algorithm can be presented as follows (Algorithm 1).

Algorithm 1 Improved $ϵ-JADE$

1:  Initialization: $NP$ = 50; 2:  Generate $NP$ individuals randomly; 3:  Evaluate the objective function and constraint violation for each individual; 4:  Set G = 0; $FES$ = 0; 5:  while $FES<MaxFES$ do 6:     $S_F$ = []; $S_{CR}$ = []; 7:     $G=G+1$; 8:     for each individual i in the population $NP$ do 9:          $C{R}_i$ = $rand{c}_i$(${\mu }_{CR}$,0.1); $F_i$ = $rand{n}_i$(${\mu }_F$,0.1); 10:          Use DE/current-to-pbest/1 mutation operator to generate the mutation vector. 11:          Use crossover ranking technique to assign the crossover control parameter. 12:          Use exponential crossover to generate the trial vector. 13:          Apply Formula (7) to compare the trial vector with the base vector and choose the better one to survive into the next generation. 14:     end for 15:     Update the pbest individuals. 16:     Apply the Formula (8) to update the $\epsilon$ value 17:     Set $FES=FES+NP$ 18:  end while

3. Problem Formulation

This section will introduce the mathematical model of the OPF problems. The variables, constraints and objectives will be presented.

3.1. Problem Formulation

The OPF problems aim to optimize the objective function within the feasible region that satisfies the constraints. The general model of the constrained optimization problem could be formulated as follows:

$$\begin{array}{c}\hfill MinimizeF\left(x\right)\\ \hfill Subjectto\\ \hfill g_i\left(x\right)=0,i=1,2,3,...,m\\ \hfill h_j\left(x\right)\le 0,j=1,2,3,...,n\end{array}$$

(9)

where F is the objective function, x is the decision variable, $g_i$ indicates the equality constraints, $h_j$ is the inequality constraints. and m and n are the total number of equality and inequality.

3.1.1. Decision Variables

In OPF problems, the decision variables can be divided into dependent and independent variables. The independent variables can be presented as follows:

$$\begin{array}{c}\hfill U={[P_G,V_G,Q_C,T_{t}ap]}_{1\times d}\\ \hfill P_G={[P_{G_1},P_{G_2},...,P_{G_{(}{N}_G-1)}]}_{1\times N_G}i\ne slack\\ \hfill V_G={[V_{G_1},V_{G_2},...,V_{G_{(}{N}_G)}]}_{1\times N_G}\\ \hfill Q_C={[Q_{C_1},Q_{C_2},...,Q_{C_{N_C}}]}_{1\times N_C}\\ \hfill T_{t}ap={[T_{{tap}_1},T_{{tap}_2},...,T_{{tap}_{N_T}}]}_{1\times N_T}\\ \hfill d=(N_G-1)+N_G+N_C+N_T\end{array}$$

(10)

where $P_G$ denotes the output active power of the generator. $N_G$ is the number of generators. $V_G$ represents the voltage of the generation bus. $Q_C$ is the injected reactive power of the shunt compensator. $N_C$ is the number of the shunt compensators. $T_{t}ap$ denotes the tap setting of the transformers. $NT$ represents the number of the transformers. d is the dimension of the decision variables. The dependent variables include the generated power of the slack bus, voltage of the load bus, load bused number, the reactive power output generators, the apparent power flow in transmission line, and the number of transmission lines. The independent variables include the output active power of the generator, generator number, voltage of the generation bus, injected reactive power of shunt compensators, the number of shunt compensator, the tap setting of transformers and the number of the transformers. The dependent variables can be given as follows:

$$X=[P_{G_0},V_{L,1},...,V_{L,NPQ},Q_{G,1},...,Q_{G,NPV},S_{TL,1},...,S_{TL,NTL}]$$

(11)

3.1.2. Objective Functions

Six single objective functions of OPF and four multi-objective functions were studied in this part. The multi-objective functions are transformed into single objectives with weighted factors. The multi-objective method can be used to tackle the objectives [20].

Case 1: Fuel cost minimization

The fuel cost ($/h) is the most commonly used objective in OPF problems. The quadratic function is employed to model the generation cost.

The formulas can be given as follows:

$$F_1\left(X\right)=\sum _{i=1}^{NG}(a_i·P_{G_i}^2+b_i·P_{G_i}^2+c_i)$$

(12)

where the $a_i$, $b_i$ and $c_i$ are the cost coefficients of the i-th generator.

Case 2: Fuel cost with valve point-effect minimization

The fuel cost with valve point-effect and prohibited zones is used to describe the generation cost. It consists of two parts. One of them is the total fuel cost $F_1$, and the other is a sinusoidal function. A sinusoidal function is used to simulate the valve effect.

$$F_C=\sum _{i=1}^{NG}(a_i·P_{G_i}^2+b_i·P_{G_i}^2+c_i)+\left|d_i·sin(e_i·(P_{G_i}^{min}-P_{G_i}))\right|$$

(13)

where the $d_i$ and $e_i$ are the fuel cost coefficients of the i-th generator with valve-point effects. The input flow of the steam is adjusted by the generation units to control the output flow by different valves.

Case 3: Voltage stability minimization

The voltage is an important index in power system stability, which should be varied within a suitable region. Hence, the load bus L-index is used, whose value should be within the range of 0 to 1. The formula can be given as follows:

$$\begin{array}{c}\hfill L_j=\left|1-\sum _{i=1}^{NG}{F}_{ij}·V_i/V_j\right|,j=1,...NL.\\ \hfill F_{V-stability}=max\left(L_j\right),j=1,...NL.\end{array}$$

(14)

where $V_i$ and represent the voltage of the i-th generator and j-th P-Q bus, respectively; $F_{ij}$ is calculated from the X-matrix.

Case 4: Emission minimization

The emission in the power system is an important environmental index that is highly related to global warming and carbon emission issues. Hence, the emission in this case is formulated with the following formula:

$$F_E=\sum _{i=1}^{NG}{\alpha }_i+{\beta }_i·P_{G_i}+{\gamma }_i·P_{G_i}^2+{\delta }_i·e^{{\varphi }_i{P}_{G_i}}$$

(15)

where ${\alpha }_i,{\beta }_i,{\gamma }_i,{\delta }_i$ and ${\varphi }_i$ represent the i-th generator emission coefficients, and their values are provided in

Table 1. Cost and emission coefficients of generators for the IEEE 30-bus system.

Generator	Bus	a	b	c	d	e	alpha	beta	gamma	w	mu
G1	1	0	2	0.00375	18	0.037	4.091	−5.554	6.49	0.0002	2.857
G2	2	0	1.75	0.0175	16	0.038	2.543	−6.047	5.638	0.0005	3.333
G3	5	0	1	0.0625	14	0.04	4.258	−5.094	4.586	0.000001	8
G4	8	0	3.25	0.00834	12	0.045	5.326	−3.55	3.38	0.002	2
G5	11	0	3	0.025	13	0.042	4.258	−5.094	4.586	0.000001	8
G6	13	0	3	0.025	13.5	0.041	6.131	−5.555	5.151	0.00001	6.667

.

Case 5: Real power loss minimization

Power loss in transmission systems is inevitable due to the intrinsic resistance of the wires. Therefore, the active power loss is worth considering. The mathematical model of this case is given as follows:

$$F_{Loss}=\sum _{i=1}^{nl}\sum _{i\ne j}^{nl}{G}_{ij}·[V_i^2+V_j^2-2V_i·V_j·cos({\delta }_i-{\delta }_j)]$$

(16)

Case 6: emission and generation cost minimization

The emission cost and generation cost are both considered in this case. The emission and generation costs are formulated as follows:

$$F_{VD}=\sum _{i=1}^{NL}(V_{L_P}-1.0)$$

(17)

Case 7: Fuel cost and real power loss minimization

This multi-objective case is converted to a single objective by multiplying a weight factor by one of the objectives. The problem can be formulated as follows:

$$F_1\left(X\right)=\sum _{i=1}^{NG}(a_i·P_{G_i}^2+b_i·P_{G_i}^2+c_i)+{\lambda }_P·F_{Loss}$$

(18)

where $F_{Loss}$ is calculated according to Case 5 and the factor of ${\lambda }_P$ is set as 40.

Case 8: Fuel cost and voltage deviation minimization

Voltage deviation is a measure of voltage quality in the network. The index of deviation is also vital from a security aspect. The indicator is formulated as the cumulative deviation of the voltages of all load buses (PQ buses) in the network from the nominal value of unity. The formula is as follows

$$F_1\left(X\right)=\sum _{i=1}^{NG}(a_i·P_{G_i}^2+b_i·P_{G_i}^2+c_i)+{\lambda }_{V}D·VD$$

(19)

where ${\lambda }_{V}D$ is a weighted factor set as 100.

Case 9: Fuel cost and enhancement of voltage stability minimization

The objective function is to minimize the weighted sum of the fuel cost and enhance the voltage stability of the system. The multiple objectives are converted to a single objective, as follows.

$$F_1\left(X\right)=\sum _{i=1}^{NG}(a_i·P_{G_i}^2+b_i·P_{G_i}^2+c_i)+{\lambda }_L·L_{max}$$

(20)

where ${\Lambda }_L$ is the weighted factor set as 100.

Case 10: Fuel cost, emission, voltage deviation and power losses minimization

Four objectives, which are fuel cost, emission, voltage deviation and power losses, are considered in this case.

$$F_1\left(X\right)=\sum _{i=1}^{NG}(a_i·P_{G_i}^2+b_i·P_{G_i}^2+c_i)+{\lambda }_E·F_E+{\lambda }_{V}D·F_{VD}+{\lambda }_P·F_{Loss}$$

(21)

where three weighted factors are set as ${\lambda }_E=19$, ${\lambda }_{V}D=21$ and ${\lambda }_P=22$, which can be referred to [21].

3.1.3. Constraints

There are various types of constraints in OPF problems. The active power should be equal to the reactive power, which are equality constraints.

$$\begin{array}{c}\hfill P_{G_i}-P_{D_i}-V_i·\sum _{j=1}^{NB}{V}_j·[G_{ij}cos({\delta }_i-{\delta }_j)+B_{ij}sin({\delta }_i-{\delta }_j)]=0,i=1,...,NB\\ \hfill Q_{G_i}-Q_{D_i}-V_i·\sum _{j=1}^{NB}{V}_j·[G_{ij}cos({\delta }_i-{\delta }_j)-B_{ij}sin({\delta }_i-{\delta }_j)]=0,i=1,...,NB\end{array}$$

(22)

where $P_{D_i}$ represents the active load. $Q_{D_i}$ is the reactive load and ${\delta }_i$ is the i-th bus voltage angle. $NB$ is the number of the buses. $G_{ij}$ and $B_{ij}$ are the transfer conductance and the susceptance between buses i and j, respectively. There are four inequality constraints, which can be presented as follows:

(1)

Generator constraints:

$$\begin{array}{c}\hfill P_{G_i}^{min}\le P_{G_i}\le P_{G_i}^{max},i=1,..,NG\\ \hfill Q_{G_i}^{min}\le Q_{G_i}\le Q_{G_i}^{max},i=1,..,NG\\ \hfill V_{G_i}^{min}\le V_{G_i}\le V_{G_i}^{max},i=1,..,NG\end{array}$$

(23)

where the i-th bus generator active power $P_{G_i}$, reactive power output $Q_{G_i}$ and voltage magnitude at the i-th generator bus $V_{G_i}$ are generated within their lower and upper bounds.

(2)

Shunt compensator constraints:

$$Q_{C_j}^{min}\le Q_{C_j}\le Q_{C_j}^{max},j=1,..,NC$$

(24)

where the shunt compensator at the j-th bus $Q_{C_j}$ should lie in its lower and upper limits.

(3)

Transformer constraints:

$$T_k^{min}\le T_k\le T_k^{max},k=1,..,NT$$

(25)

where the k-th branch transformer tap $T_k$ is within its lower and upper limits.

(4)

Security constraints:

$$\begin{array}{c}\hfill V_{L_m}^{min}\le V_{L_m}\le V_{L_m}^{max},k=1,..,NL\\ \hfill S_{L_n}\le S_{L_n}^{max},n=1,...,nl\end{array}$$

(26)

where the voltage magnitude at the m-th load bus $V_{L_m}$ and n-th line loading should follow its lower and upper bounds.

Experimental Results

The IEEE 30-bus test system is used to test the effectiveness of the algorithm. The system has 6 bus generators, and 24 load buses. The lower bounds of the voltage magnitude is 0.95 p.u. The upper bounds of the PV buses and PQ buses are 1.1 p.u. and 1.05 p.u., respectively. The lower and upper bound of transformer tappings are 0.9 and 1.1 p.u., respectively. The proposed I$ϵ$JADE algorithm is compared with state-of-the-art DEs, which are ECHT-DE [22], SF-DE [22], SP-DE [22] and ACDE [1].

3.2. Results of the OPF Problems

In this subsection, the proposed algorithm is compared with ECHT-DE. SF-DE, SP-DE and ACDE. The experimental results are shown in

Table 2. Experimental results of the I$ϵ$JADE and the state-of-the-art algorithms.

Case	Algorithm	Min	Max	Mean	Std
Case1	ECHT-DE	800.4148	800.4258	800.4206	0.0026
	SF-DE	800.4131	800.4192	800.4151	0.0015
	SP-DE	800.4293	800.4684	800.4413	0.0100
	ACDE	800.4113	800.4176	800.4133	0.0015
	I$ϵ$-JADE	800.4115	800.4229	800.4135	0.0021
Case2	ECHT-DE	832.1356	832.2239	832.1811	0.0222
	SF-DE	832.0882	832.1291	832.1056	0.0105
	SP-DE	832.4813	832.8760	832.6550	0.0963
	ACDE	832.0722	832.3941	832.0957	0.0581
	I$ϵ$JADE	832.0698	832.1225	832.0809	0.0109
Case3	ECHT-DE	0.1363	0.1372	0.1369	0.0002
	SF-DE	0.1367	0.1370	0.1369	0.0001
	SP-DE	0.1374	0.1386	0.1378	0.0002
	ACDE	0.1364	0.1368	0.1366	0.0001
	I$ϵ$JADE	0.1364	0.1367	0.1365	0.0001
Case4	ECHT-DE	0.2048	0.2048	0.2048	0.0000
	SF-DE	0.2048	0.2048	0.2048	0.0000
	SP-DE	0.2048	0.2048	0.2048	0.0000
	ACDE	0.2048	0.2048	0.2048	0.0000
	I$ϵ$JADE	0.2048	0.2048	0.2048	0.0000
Case5	ECHT-DE	3.0850	3.0871	3.0858	0.0005
	SF-DE	3.0845	3.0857	3.0849	0.0003
	SP-DE	3.0844	3.0854	3.0848	0.0003
	ACDE	3.0840	3.0862	3.0845	0.0005
	I$ϵ$JADE	3.0840	3.0851	3.0844	0.0003
Case6	ECHT-DE	0.0878	0.0916	0.0893	0.0009
	SF-DE	0.0867	0.0890	0.0880	0.0007
	SP-DE	0.0867	0.0892	0.0877	0.0007
	ACDE	0.0856	0.0878	0.0865	0.0007
	I$ϵ$JADE	0.0856	0.0884	0.0863	0.0007
Case7	ECHT-DE	1040.1510	1040.2330	1040.1810	0.0213
	SF-DE	1040.1250	1040.1620	1040.1400	0.0096
	SP-DE	1040.1340	1040.3370	1040.2390	0.0444
	ACDE	1040.1133	1040.1891	1040.1268	0.0177
	I$ϵ$JADE	1040.1127	1040.1642	1040.1245	0.0115
Case8	ECHT-DE	813.1742	813.4095	814.2470	0.0490
	SF-DE	813.1956	813.3376	813.2585	0.0444
	SP-DE	813.1959	813.2643	813.2306	0.0181
	ACDE	813.1100	813.5334	813.1379	0.0805
	I$ϵ$JADE	813.1099	813.5583	813.1462	0.0846
Case9	ECHT-DE	814.1708	814.2001	814.1843	0.0075
	SF-DE	814.1649	814.1956	814.1767	0.0063
	SP-DE	814.1841	814.2273	814.2017	0.0121
	ACDE	814.1588	814.2957	814.1897	0.0305
	I$ϵ$JADE	814.1588	814.2162	814.1746	0.0118
Case10	ECHT-DE	964.1331	964.1564	964.1437	0.0061
	SF-DE	964.1254	964.1418	964.1307	0.0038
	SP-DE	964.1234	964.1399	964.1276	0.0034
	ACDE	964.1179	964.1493	964.1252	0.0083
	I$ϵ$JADE	964.1176	964.1380	964.1227	0.0050

. It is worth mentioning that the control parameters of the proposed algorithm are the same as ACDE. The biggest difference between ACDE and I$ϵ$-JADE is the constraint handling technique. ACDE uses the SF method rather than the $ϵ$ constraint method in I$ϵ$-JADE.The maximum, minimum and average values of thirty independent runs are shown in Table 2. The best results among all the algorithms are marked in boldface.

• For case 1, the simulation results show that ACDE has a better Min values, Max values and Std values than the improved $ϵ$JADE. Improved $ϵ$-JADE presents better mean values in this case, which indicates that the effectiveness of the proposed algorithm.
• For case 2, case 3, case 5, case 6 and case 10, the proposed improved $ϵ$-JADE algorithm exhibits a competitive performance on the Min, Max and Mean values when compared with the other four state-of-the-art DE algorithms;
• For case 4, all compared DE algorithms provide the same competitive results.
• For case 8, ACDE and SP-DE achieve the best Mean value, while the improved $ϵ$JADE demonstrates a remarkable performance in terms of the Min value;
• With regard to case 7 and case 9, the proposed improved $ϵ$JADE demonstrates better results in terms of the Mean and Min values.

In addition, the comparison results between the ACDE and improved $ϵ$JADE are shown in the box plot in Figure 1. In the first four cases, the improved $ϵ$JADE and ACDE are equally good. Meanwhile, in case 5 and case 6, the improved $ϵ$JADE demonstrates a better performance.

The convergence curves of ACDE and the improved $ϵ$ JADE ($ϵ$ JADE is used for short in the legend) are shown in Figure 2. The figure shows that the improved $ϵ$ JADE has a competitive convergence rate with ACDE. As mentioned above, using the $ϵ$ method will lead to a slow convergence speed towards the feasible region compared with SF. For constrained optimization problems with a relatively small feasible region, the $ϵ$ method is more effective in guiding the population to move toward the feasible region. By considering both target values and constraints, the proposed method will not lead to a significant decrease in the convergence speed.

Based on the comparison results of five DE variants, it is found that the performance of the algorithm, especially the robustness, is significantly improved by the $ϵ$ method.

From the Figure 3, the load bus voltages of different load buses in all the six cases are within bound, which means that the solutions are all feasible. We can observe that the load bus voltage of different load buses follows the same trend, which can help solve other problems in this series.

Comparison between Improved $ϵ$JADE and ACDE

The convergence curves of the improved $ϵ$JADE and ACDE in the first six cases are given in Figure 2.

Compared with the ACDE, it can be observed that the proposed improved $ϵ$JADE has a competitive performance with ACDE. The convergence speed of the proposed algorithm is faster than that of ACDE in most cases.

From the boxplots in Figure 1, it can be observed that the proposed algorithm obtains more stable and robust results than ACDE, which illustrates the effectiveness of the improved algorithm.

4. Discussion

Compared with state-of-the-art algorithms in dealing with OPF problems, it can be concluded that the improved $ϵ$ method with the adaptive differential evolution can achieve competitive results. In dealing with the constrained optimization problems, the algorithm is important in searching for the optimal and constraint handling method. The adaptive differential evolution can be applied in solving the constrained optimization problems without extra computations. It is still more efficient in finding the feasible solutions than the classical differential evolution. The proposed selection based on the constraint violation is simple yet effective in improving the algorithm. The $ϵ$ method has shown a great performance in complex constrained optimization problems [23]; meanwhile, it is effective without losing the efficiency through the experimental results in OPF problems. Compared with the simple feasible rules methods, the improved $ϵ$ method can be more suitable in dealing with the complex OPF problems.

5. Conclusions

In this paper, an improved $ϵ$ method based on the adaptive differential evolution is utilized to solve the optimal flow problems. The improved $ϵ$ method can help the algorithm move toward the feasible region, and the improved algorithm is efficient in searching for the feasible global optimum. The effectiveness of the proposed algorithm is tested on the IEEE-30 buses series benchmark functions. Compared with the state-of-the-art algorithms, the performance of the proposed algorithm is competitive in terms of the convergence speed and precision.

In the future, the proposed algorithms could be used to solve more complex optimal flow problems. More effective constraint handling techniques could be combined with the improved adaptive differential evolution algorithms in dealing with the complex constrained optimization problems. It is also promising to implement the machine-learning-based parameter-setting methods within the algorithm rather than fine tuning the control parameters by experiments.