1.   Introduction

Optimizing reactive power flow is an operation, in which the structure parameters and load conditions of the power system are given and under the premise to meet all the specified constraints, one or more performance indicators of the power system are optimized by regulating some control variables. Reactive power is a key factor affecting the quality of power system voltage, so it is important to improve the quality of power system voltage by optimizing reactive power flow.

Optimal reactive power flow (ORPF) is a multi-variable, multi-constraint, discontinuous, and nonlinear optimization problem. It involves continuous variables and discrete variables, which make it difficult to solve. Over the years, ORPF has been a popular issue for researchers in both domestic and abroad. Many effective methods, such as genetic algorithms, differential evolution algorithms, particle swarm algorithms, etc., have been proposed. These optimization algorithms only need one adaptive function or one indicator, and have global convergence performance, thus they are being employed in various areas of power system research and its applications.

In ORPF, Genetic algorithms are used widely [1], [2]. GA needs less information and the corresponding model is very simple. It is suitable for multi-constraints and variables, nonlinear problems, and is applied widely. Differential evolution algorithm is an important branch of genetic algorithm, which has been proved to be an effective solution for ORPF. DE is an intelligent optimization algorithm based on the population theory, it preserves the global search strategy based on population theory and takes simple operation such as selection, cross and variation as the population evolution strategy against the diversity of candidate solution, and its principle is easy to understand. DE does not have the problems to provide feature information, and it can be applied to some optimization problems which the regular mathematical programming methods could not solve, and it has good generality. Therefore, DE is suitable for some optimization problems in complex environment which cannot be solved by conventional mathematical method, such as ORPF. Varadarajan et al. [3] builds a nonlinear model for ORPF with mixed continuous and discrete variables, then uses DE to solve ORPF, and compares DE with traditional sequence quadratic program (SQP). The results show that DE has a significant advantage. Basu [4] uses DE, GA and evolutionary programming (EP) to solve ORPF problem with flexible ac transmission systems (FACTS) devices, makes simulation on IEEE-30 bus system, and analyze results of DE, GA and EP. The results show that DE is more suitable for ORPF. But DE is prone to "premature convergence" problem, and slow convergence speed, so it is difficult to achieve the effective control on the real-time system [5]. Different algorithms have different advantages and disadvantages, so researchers always combine the advantages of every algorithm and propose a new hybrid algorithm, which are more effective ORPF. Zhao et al. [6] combines particle swarm optimization (PSO) algorithm with multi-agent system, and makes simulation on IEEE 30bus system and IEEE 118-bus system. The results show that the optimization time and precision performance of improved algorithm are better than original one. Fang et al. [7] improves the traditional genetic operators of genetic algorithm, coding and termination conditions, etc., and then makes simulation on IEEE 14-bus system, the results show that the new algorithm is better. Wang et al. [8] proposes a hybrid algorithm, improves the convergence by simulated annealing method, and achieves higher precision and accuracy by tabu search algorithm. Liu et al. [9] puts forward a kind of hybrid intelligent algorithm combining the artificial colony evolution thought and particle swarm optimization algorithm, simulation results show that hybrid algorithm effectively overcomes the defect of particle swarm optimization algorithm. Ideas of quantum superposition and quantum revolving door is introduced into particle swarm optimization (PSO) algorithm [10], which improves the convergence speed. Li [11] improves differential evolution algorithm by artificial colony algorithm search ideas, and does simulations to validate the performance of the proposed algorithm on 14-bus system and 30-bus system. The results show that the algorithm reduces the population size and convergence time. So this paper combines quantum encoding and artificial bee colony algorithm (ABC) with DE, by improving the individual diversity and search ability of DE.

As more and more distributed generations (DGs) are introduced into the power system, there is a certain impact on power flow and voltage of distribution network, which affects the robustness and effectiveness of many traditional algorithms. Enlightened by the above-mentioned intelligent hybrid algorithms, from the view of traditional differential evolution algorithm premature convergence and the shortcomings of poor local search ability, this paper improves the basic differential evolution algorithm through artificial colony algorithm and the quantum computing ideas. Then this paper proposes an improved quantum differential evolution algorithm (IQDE) and applies it to OPRF problem, which significantly improves the convergence rate of the power system reactive power optimization and the optimization performance.

The paper is organized as follows. The mathematical model of ORPF problem is first reviewed in Section 2. Then, Section 3 introduces the principles of the proposed hybrid algorithm and how to introduce quantum ideas into DE and ABC. In Section 4, the effectiveness of IQDE is verified by simulations on the IEEE 14-bus and 30-bus system including DGs. In the last, Section 5 concludes the paper.

2.   Problem Formulation

ORPF is a problem which makes the active power network losses reach the intended target of power system, and makes voltage quality and reactive power compensation meet the various constraints such as security and stability [12]-[14]. In this paper, from the perspective of economy, the selected objective function is to minimize active power network losses, where the control variables include: generator voltage, tap position of transformer, and capacity of reactive power compensator.

1) Objective Function

where $n$ denotes the total branch numbers of system, ${G_k}(i, j)$ denotes the conductance of branch $k$ which connects node $i$ with node $j$ , $u_i$ and $u_j$ denote the voltage of node $i$ and $j$ , respectively. ${\theta_i}$ and ${\theta_j}$ denotes the phase angle of node $ i$ and $ j$ , respectively.

2) Equality Constraints

Equality constraint is the power flow equation constraint, which the reactive power and active power injected in any node $i$ should meet:

where $i \in N$ , $N$ denotes the sets of all nodes in power distribution system, $P_{i}$ and $Q_{i}$ denote the active power and reactive power of node $i$ , respectively. $u_{i}$ and $u_{j}$ denote the voltage of node $i$ and $j$ , $G_{ij}$ , $B_{ij}$ , respectively. $\delta_{ij}$ denote the conductance, susceptance and voltage phase angle difference of node $i$ and $j$ , respectively. $N_{i}$ denotes the node sets of the other end to which all the branches leave from node $i$ .

3) Inequality Constraints

Inequality constraints are bound variables, including the state variables bound constraints and control variables bound constraints.

Bound constraints of state variables include:

Bound constraints of control variables include:

where $Q_{Ci}$ denotes the reactive power output of generator node $G_{i}$ , $U_{i}$ denotes the voltage of load node $i$ . $U_{Gi}$ denotes the voltage of generator node $G_{i}$ , $T_{i}$ denotes the ratio of on-load tap changer, $Q_{Ci}$ denotes the reactive power compensation capacity of reactive power device $C_{i}$ .

3.   The Proposed Hybrid Algorithm

3.1   Related Works

3.1.1   Differential Evolution Algorithm

The basic differential evolution algorithm (DE) bases on the swarm intelligence theory raised by Rainer Storn and Kenneth Price in 1997 [15]. It extracts differential information from the current population, and guides the further search to solve some optimization problems in complex environment which cannot be solved by conventional mathematical programming methods. DE algorithm is simple and has common control parameters and good robustness. But there are also some deficiencies in DE algorithm such as premature convergence, poor local search ability, etc. This paper improves it through artificial bee colony algorithm and the quantum computing ideas, in order to apply it in reactive power optimization problem.

3.1.2   Artificial Bee Colony Algorithm

Artificial bee colony algorithm (ABC) is a novel evolutionary algorithm proposed in 2005, with fewer parameter settings, fast convergence and high convergence accuracy, which is applied in the field of power system. The basic concept includes food sources and foraging bees, food sources is the solution of optimization problem, foraging bees is divided into three types of bees, which are employed bees, onlooker bees and scout bees [16]. Artificial bee colony system combines local search methods with global search methods, employed and onlooker bees perform local search, onlooker bees and scout bees perform global search [17].

Considering the shortcomings of differential evolution algorithm, we can use the swarm behavior of artificial bee colony algorithm to improve it:

a) Onlooker Bee Operation

In ABC algorithm, onlooker bee determines its own search range according to foraging information of employed bee, the idea can be applied into the differential evolution algorithm in which individual evolves with the same probability. If swarm accelerating evolution operation is combined with DE algorithm, better evolved individual can offer more opportunities for local search, so as to improve the local search ability of DE algorithm.

b) Scout Bee Operation

If artificial scout bee operation is combined with DE algorithm, the individuals which don't improve after continuous evolution will be abandoned with random individuals instead, so as to improve the global search ability of DE algorithm.

3.1.3   Quantum Computing

In quantum computing, the smallest unit of information is qubits, which can be in state $|0\rangle$ , state $|1\rangle$ and any superposition state $|0\rangle$ with $|1\rangle$ . Its state can be expressed as:

In (6), $\alpha $ , $\beta $ is a pair of complex number, which denotes respectively as probability amplitude of $|0\rangle$ with $|1\rangle$ , and satisfies:

A qubit can be defined as by probability amplitude. Similarly, $m $ qubits can be defined as in which, $|\alpha|^{2}+|\beta|^{2}=1$ , . Because superposition state system can be described by quantum system, the evolution algorithm based quantum encoding has better population diversity than the traditional DE [18].

3.2   Improved Quantum Differential Evolution Algorithm

This paper presents an improved quantum differential evolution algorithm (IQDE) based on the idea of improving the DE algorithm by quantum computation and ABC algorithm. As DE needs large population size to avoid premature convergence and poor local search performance, we are looking for a suitable algorithm to improve it. ABC algorithm is widely used in the field of power system with strong global and local search capabilities. Adding ABC's onlooker bee accelerating evolution behavior in DE can improve its local search performance. In addition, the hybrid algorithm adopts the quantum chromosome encoding to represent the optimization variable values, which can ensure the diversity of population, and provide better conditions for optimal search of the next step. In addition, introducing the artificial scout bee operation into DE and replacing the individual which has not been improved after continuous evolution by a random individual can guarantee quantum individuals search better quantum individuals from other areas, and enhance the capability of the hybrid algorithm to jump out of local optimal solution. The explanation of proposed algorithm is as follows, (1)-(5) is quantum operation, (6) and (7) is ABC method.

3.2.1   Quantum Initialization

In IQDE, qubit probability amplitude is represented in the form of quantum angle , state $|0\rangle$ and state $|1\rangle$ satisfy $|\mathop {\cos}\theta|^{2}$ + $|\mathop {\sin}\theta|^{2}$ = 1. So the population size is $n$ , quantum chromosome encoding whose variable dimension is $m$ can be defined as:

where, $i=1, 2, \ldots, n.~t$ represents the evolution algebra. Equation (8) can be simplified in the form of quantum angle as $[\theta_{1}|\theta_{2}|\cdots|\theta_{m}]$ . The angle $\theta_{j}^{t}$ \mbox{( $j$ = 1, 2, $\ldots$ , $m$ ; $t$ = 0)} of quantum individual is formed in [0, 2 $\pi$ ] randomly. We can see that each quantum individual in the population occupies two positions in the space, which is corresponding to the two solutions of optimization problems respectively and corresponding to the probability amplitude of state $|0\rangle$ and $|1\rangle$ , respectively.

3.2.2   Solution Space Transformation

The range of quantum state probability amplitude is [0, 1]. By setting the values of solution space variables in [ $a$ , $b$ ], we need to transform the solution space, so that each probability amplitude can correspond to the variables. Denote the variable $j$ of the generation $t$ , individual $i $ as $x_{i}^{j}$ ( $t$ ), then the quantum encoding is () $^{T}$ , the decoding process is done by using (11) and (12).

where the probability amplitude of quantum state $|0\rangle$ corresponds to $x_{ic}^{j}(t)$ , the probability amplitude of quantum $|1\rangle$ corresponds to $x_{is}^{j}(t)$ .

3.2.3   Quantum Mutation Operation

According to the basic differential evolution algorithm, we make variation of each quantum individual correspond to each individual $q_{i}^{t}$ , its phase is:

where $r_{1}, r_{2}, r_{3} \in $ {1, 2, $\ldots$ , 2 $n$ }, $r_1\neq r_2 \neq r_3 \neq i$ , $F \in [0,1]$ . This step generates $n$ variation individuals in total.

3.2.4   Quantum Crossover Operation

Generate a random integer $j_{\rm rand} \in$ {1, 2, $\ldots$ , $m$ }, then get the phase of the test individual:

where $j\in{1, 2, \ldots, m}, CR\in[0,1]$ .

3.2.5   Quantum Selection Operation

Before the selection operation, we calculate the fitness value of the individual firstly. By comparing the fitness value of test individuals with that of current generation quantum individuals, we select the optimal individuals for inclusion into the next generation.

To calculate the fitness value, we need to carry on the solution space transformation firstly. Convert quantum individual $q_{ic}^{t}$ and $q_{is}^{t}$ to solution space variables $x_{ic}^{j}$ ( $t$ ) and $x_{is}^{j}$ ( $t$ ) by (11) and (12). The test individual phase $u_{ij}^{t+1}$ ( $t$ ) are taken cosine and sine, respectively, and converted to $y_{ic}^{j}$ ( $t$ ) and $y_{is}^{j}$ ( $t$ ) after after the solution space transformation. Calculate the fitness value of the four groups of solution space variables, and take the individual with minimum fitness value (its phase is marked as $\theta_{\rm best}$ ) as the better individual for inclusion into the next evolution or the next iteration.

The individual diversity can increase via quantum operation as (1)-(5).

3.2.6   Artificial Bee Colony Accelerated Evolution Operation

The bee of ABC algorithm requires food, the more the better, and the objective of DE to solve the problem of ORPF is to minimize the active power losses, so it is necessary to do reverse adjustment for the fitness value of DE. Introduce a fitness distribution:

where .

According to the individual fitness distribution, we can compute the times of each individual which participate in the accelerating evolution in each round of iteration:

Then do basic differential evolution operation for each individual $N_{i}$ times, and finally produce the best individual for inclusion into the next iteration.

3.2.7   Artificial Bee Colony Scout Operation

After several simulations, we define the maximum number of evolutions as $N_{\rm max}$ = I $_{-}$ itermax/5 by experience. I $_{-}$ itermax is the maximum number of iterations. If an individual has not improved after more than $N_{\rm max}$ times evolution, eliminate it and replace it with a random individual according to (8).

3.3   Algorithm Process

Fig. 1 is the flowchart of IQDE hybrid algorithm, the process is as follows:

图 1  The flowchart of IQDE hybrid algorithm.

Fig. 1  The flowchart of IQDE hybrid algorithm.

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Step 1: Quantum initialization: generate the initial population containing $N$ quantum chromosomes according to (8). Each quantum individual in the population is respectively corresponding to two solutions of the optimization space, the population size is $n$ , the individual vector dimension is $m$ and the number of the generation is $t$ . Set the value of scaling factor $F$ .

Step 2: Quantum operation of differential evolution: include quantum mutation operation, quantum crossover operation and quantum selection operation, as (3)-(5) given in Section 2. Do the above quantum differential evolution for each individual, and choose a better individual (fitness value smaller) for the next evolution. In the quantum selection operation, we should do solution space transformation first to calculate the fitness value, as (11) and (12) shown. Make the original quantum chromosome and the chromosome after mutation and crossover correspond to the two solutions of the optimization problems respectively, calculate each solution's fitness value, update the optimal solution according to the principle of greedy method.

Step 3: Artificial bee colony accelerating evolution operation: compute and do reverse adjustment for the individual fitness value, then compute the individual fitness distribution. And then compute the evolution cycling number $N_{i}$ . For each individual, do quantum operation of DE $N_{i}$ times.

Step 4: Determine whether there are unimproved individuals, if individuals are still not improved after the maximum evolution times $N_{\rm max}$ , enter Step 5, otherwise Step 6.

Step 5: Artificial bee colony scout operation: eliminate the unimproved individual, replace it with a random individual according to (8).

Step 6: Determine whether the termination condition is reached: If the iteration is completed or the convergence condition is satisfied, the algorithm will save the results of optimization with the algorithm termination. Otherwise enter Step 2 for the next round of iteration.

4.   Experimental Results

4.1   IEEE 14-bus System

4.1.1   Parameter Settings of 14-bus System

DE and IQDE are applied in ORPF problem, respectively. Several simulations are done in the standard IEEE 14-bus system, then the effect of the improved algorithm is verified by comparing the performance of the two algorithms. Simulations are conducted in MATLAB R2010a, hardware environment is: 4.00 GB RAM, 2.50 GHz CPU quad-core.

The mainly involved simulation parameters refer to the parameters of DE algorithm. After several rounds of verification, we ultimately determine the scaling factor $F$ = 0.6, the cross factor $CR = 0.5$ , I $_{-}$ itermax = 100. In addition, the study compares the performance of the two algorithms when the setting of the population size is different. Repeat the simulation 30 times in the 14-bus system.

Table Ⅰ lists the amount of each control variables, in which, $SUM$ is the total number of control variables, $T$ is the number of on-load tap changer, $U$ is the number of generator bus, $Q$ is the number of compensation capacitor.

表 Ⅰ  Number of Control Variables of IEEE 14-Bus System

Table Ⅰ  Number of Control Variables of IEEE 14-Bus System

Variable	Number
T	3
U	5
Q	2
SUM	10

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Table Ⅱ is the setting of control variables of IEEE 14-bus system. The generator bus voltages are continuous variables so they cannot be represented by "step", so we denote it as "-". For state variables, the upper limit of the PQ-bus voltage is set to 1.05 p.u. (per unit), the lower limit of it is set to 0.95 p.u., constraint conditions of the reactive power output of the generator node (as a PV bus) are shown in Table Ⅲ.

表 Ⅱ  Setting of Control Variables of IEEE 14-Bus System

Table Ⅱ  Setting of Control Variables of IEEE 14-Bus System

Variable	Min	Max	Step
T	0.9	1.1	0.01
U	0.9	1.1	-
Q	0	0.18	0.06

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表 Ⅲ  Constraints of The State Variables of IEEE 14-Bus

Table Ⅲ  Constraints of The State Variables of IEEE 14-Bus

Node	Min (MVar)	Max (MVar)
1	0	10
2	-40	50
5	-40	40
8	-10	40
11	-6	24
13	-6	24

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4.1.2   The Experimental Results and Analysis of 14-bus System

Table Ⅳ compares the results of DE and IQDE. When the population size is set to 10, 20, 30, 40, 50, 60, respectively, we compare the optimized minimum active power network losses, the optimized maximum active power losses, the optimized average active power losses, the minimum convergence time, the maximum convergence time and the average convergence time. In Table Ⅳ, Loss_min, Loss_max, Loss_avg represents the minimum, the maximum and the average values of the active power network losses in the statistics with 30 times simulations, respectively. Time_min, Time_max, Time_avg represents the minimum the maximum and the average convergence time in the statistics with 30 times simulations, respectively. It can be concluded from Table Ⅳ:

表 Ⅳ  Statistics of Results for DE and IQDE (IEEE 14-Bus)

Table Ⅳ  Statistics of Results for DE and IQDE (IEEE 14-Bus)

SP	Algorithm	Lossmin	Lossmax	Lossavg	Timemin	Timemax	Timeavg
10	DE	12.3714	13.1578	12.4401	3.8977	5.3590	4.9267
	IQDE	12.3712	12.5450	12.3858	4.9335	9.0550	7.2024
20	DE	12.3713	12.6364	12.3892	8.7948	9.9069	10.6250
	IQDE	12.3712	12.4035	12.3761	7.8281	14.6796	11.2216
30	DE	12.3712	12.549	12.3876	12.569	15.7014	14.8099
	IQDE	12.3712	12.3712	12.3712	12.1678	17.7190	14.7212
40	DE	12.3712	12.4463	12.3776	16.6632	21.4070	20.2090
	IQDE	12.3712	12.3712	12.3712	15.2144	21.7840	17.3987
50	DE	12.3712	12.4319	12.3774	19.3238	26.4667	25.0056
	IQDE	12.3712	12.3712	12.3712	19.0269	26.7894	22.9460
60	DE	12.3712	12.399	12.3754	25.2242	33.3440	30.3782
	IQDE	12.3712	12.3712	12.3712	24.4536	33.4038	28.0544

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a) With the increase of population size, DE and IQDE can obtain a better solution: the smaller value of active power losses. The reason is that the increase of population size can enhance the global search capability of the two algorithms. However, the corresponding increase of the convergence time is inevitable.

b) When the size of population is equal, IQDE can obtain better solution than DE: better performance in the average losses, the minimum losses and the maximum losses.

c) When the population size is small ( $SP$ = 10, 20), the average convergence time of IQDE is more than DE. The reason is that IQDE's onlooker bee accelerating evolution operation increases times of power flow calculation, meanwhile, time of reactive power optimization is mainly consumed in power flow calculation. But as the population size continues increasing ( $SP$ = 30, 40, 50, 60), the average convergence time begins to be less than DE, the reason is that the quantum encoding has better parallelism.

d) To achieve the same optimal performance with DE, IQDE requires the smaller population size, for example, the average active power loss is 12.3754 when $SP$ = 60 in DE, while IQDE only requires $SP$ = 20, the average active power losses can reach 12.3761 closed to the former. When $SP$ = 30, the average active power loss is 12.3712.

In order to compare the performance of the two algorithms more directly, Fig. 2 shows the results of the average active power loss of the two algorithms. Fig. 3 shows the results of the average convergence time of the two algorithms. According to Figs. 2 and 3, we can draw the following conclusions: When the population size is same, the average active power losses of IQDE is less than DE and the average convergence time is gradually less than DE with the increased population size. The average active power loss of DE is 12.3754, convergence time is 30.3782 s, and the average active power loss of IQDE can reach 12.3761 when the population size is only 20, convergence time is 11.2216 s, which is much less than DE, respectively. The IQDE can obtain the same performance as DE with a smaller population size, less convergence time, hence it can save the running time of the algorithm.

图 2  The average active power losses comparison chart of 14-bus system.

Fig. 2  The average active power losses comparison chart of 14-bus system.

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图 3  The average convergence time comparison chart of 14-bus system.

Fig. 3  The average convergence time comparison chart of 14-bus system.

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Fig. 4 is the convergence curve comparison of DE and IQDE. The three curves in the figure respectively represents the convergence curve of IQDE when the population size is 20, and the convergence curve of DE when the population size is 20 and 60. It can be seen that when the population size is 20, IQDE converges to the optimal value 12.3712 with 36 times iteration, but DE converges only to 12.3752 with 100 times iteration. When the population size is up to 60, DE can only converge to the optimal value 12.3712 in 90 iterations. Obviously, IQDE has better optimization performance than DE.

图 4  Convergence curve comparison chart of 14-bus system.

Fig. 4  Convergence curve comparison chart of 14-bus system.

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Table Ⅴ lists the changes of control variables and reactive power loss before and after optimization, the generator voltage and on-load tap changer ratio are normalized values (p.u.), the unit of the capacity of the reactive power compensator is MVar, the unit of active power loss is MW.

表 Ⅴ  Control Variable Setting and P_LOSS Before and After Optimization for IEEE 14-Bus System

Table Ⅴ  Control Variable Setting and P_LOSS Before and After Optimization for IEEE 14-Bus System

	U1	U2	U3	U6	U8	T4	T5	T7	Q9	Q14	Ploss
Before	1.06	1.045	1.01	1.07	1.09	0.978	0.969	0.932	18	18	13.393
After	1.1	1.0779	1.0465	1.1	1.1	1.06	0.9	1.03	18	6	12.3712

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4.2   IEEE 30-bus System With Distributed Generators

The introduction of DGs will destroy traditional structure of power system, so reactive power optimization will be significantly affected by DGs. Traditional structure of power system is radial with a single generator, power transmission will flow from high voltage to low voltage. When DGs are introduced, they will not only output power, but also distribute power. They will change the power flow in system. Therefore, the ORPF problem of power system with DGs is more complex.

The mathematical model of ORPF problem for power system with DGs is similar to the problem in traditional system, in addition, it is still an optimization problem with multi-state variations and multi-constraints, which includes objective functions, equality constraints and inequality constraints. From the analyses made above, we can regard DGs as PQs and impose some constraints, so the state variable of reactive power output needs to be added to the original mathematical model. While the objective functions, equality constraints and other variable constraints remain the same, the reactive power output constraints of DGs are as follow:

The data of the IEEE 30-bus system are obtained from Power System Test Case Archive [EB/OL] of University of Washington. The simulation environment is same to that of IEEE 14-bus system. Nodes 1、2、5、8、11、13 are connected to generators. Active power, reactive power and voltage adopt standard per unit (p.u.).

The number of control variable and the total number of control variables in initial condition are respectively given in Table Ⅵ, in which $T$ represents the number of on-load tap changer, $U$ represents the number of generator node, and $Q$ represents the number of reactive power compensator.

表 Ⅵ  Number of Control Variables of IEEE 30-Bus System

Table Ⅵ  Number of Control Variables of IEEE 30-Bus System

Variable	Number
T	4
U	6
Q	2
SUM	12

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Table Ⅶ lists the limits of control variables and steps of discrete variable. $Q_{10}$ refers to the capacity of reactive power compensation provided by the reactive power compensator at node 10. Generator node voltage is a continuous variable and therefore cannot be expressed in steps, indicated by "-". For state variables, the upper and lower limits of PQ-bus voltages were set to 1.05 and 0.95 p.u., respectively, and the upper and lower reactive power constraint of PV node are listed in Table Ⅷ.

表 Ⅶ  Setting of Control Variables of IEEE 30-Bus System

Table Ⅶ  Setting of Control Variables of IEEE 30-Bus System

Variable	Minimum	Maximum	Step
T	0.9	1.1	0.02
U	0.9	1.1	-
Q9	0	0.2	0.05
Q24	0	0.04	0.01

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表 Ⅷ  Constraints of the State Variables of 30-Bus

Table Ⅷ  Constraints of the State Variables of 30-Bus

Node	Min (MVar)	Max (MVar)
1	0	10
2	-40	50
5	-40	40
8	-10	40
11	-6	24
13	-6	24

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Fig. 5 is 30-bus system with DGs. There are 4 nodes: No.9, No.19, No.24, and No.26, which are connected DGs. We can see reactive and active power output of DGs from Table Ⅸ, $Q_{\rm out}$ is reactive output, $Q_{\rm up / low}$ is the upper/lower limit of reactive power. $P_{\rm out }$ is active power output, is the upper/lower limit of active power. Eventually, 30 times simulations were executed respectively. And in the simulations the basic parameters are set as: $F$ = 0.6, $CR$ = 0.5, the iteration number is 100. According to the results in section 4.1, reactive power optimization has obtained the best result and there is no need to increase the population size when the generation is 30. Therefore, setting $SP$ = 36 and doing simulations 30 times, the specific optimization results are shown in Table Ⅹ and Table Ⅺ.

图 5  30-bus system with DGs.

Fig. 5  30-bus system with DGs.

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表 Ⅸ  Statistics of Results for DE and IQDE (IEEE 14-Bus)

Table Ⅸ  Statistics of Results for DE and IQDE (IEEE 14-Bus)

Node	Qout(MVar)	Qlow	Qup	Pout (MW)	Plow	Pup
9	0.0137	-0.012	0.025	0.15	0.11	0.19
19	0.0554	-0.013	0.0689	0.25	0.1	0.45
24	0.01	-0.0069	0.02	0.09	0.05	0.15
26	0.0425	-0.15	0.0638	0.185	0.124	0.32

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表 Ⅹ  Statistics of Trial Results for IQDE in 30-Bus System Containing DGS

Table Ⅹ  Statistics of Trial Results for IQDE in 30-Bus System Containing DGS

No.	Ploss	Time	No.	Ploss	Time	No.	Ploss	Time
1	16.2163	24.2024	11	16.2163	29.7159	21	16.2164	28.0426
2	16.2163	25.9639	12	16.2163	29.4342	22	16.2163	23.9344
3	16.2164	34.057	13	16.2163	27.2983	23	16.2166	33.3986
4	16.2163	28.0157	14	16.2163	29.5571	24	16.2163	28.6112
5	16.2163	30.1713	15	16.2166	34.6243	25	16.2164	23.7194
6	16.2163	26.3804	16	16.2163	30.1448	26	16.2163	24.2563
7	16.2165	31.7187	17	16.2164	25.1229	27	16.2164	32.3472
8	16.2163	29.4458	18	16.2163	25.7156	28	16.2163	24.8397
9	16.2163	25.2483	19	16.2164	33.2381	29	16.2164	28.5767
10	16.2165	33.7229	20	16.2163	25.9665	30	16.2166	33.8194

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表 Ⅺ  Statistics of Trial Results (IEEE 30-Bus)

Table Ⅺ  Statistics of Trial Results (IEEE 30-Bus)

Results	Average	Min	Max
Ploss	16.2164	16.2163	16.2166
Time	28.7097	23.7194	34.6243

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Fig. 6 shows a comparison about the convergence curve between IQDE and DE, which clearly shows the superiority of IQDE over DE. In the case that $SP$ = 36 and iteration number is 38, IQDE is convergent to the optimum point 16.2136. While DE is convergent to the optimum point 16.7488 and the iteration number is 100. Furthermore, in the case that $SP$ = 72 and iteration number is 100, DE is convergent to the optimum point 16.2528. Obviously, IQDE has better optimization results than DE.

图 6  The convergence curve of IQDE and DE (30-bus system containing DGs).

Fig. 6  The convergence curve of IQDE and DE (30-bus system containing DGs).

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Table Ⅻ lists all the control variables before and after optimization, in which $P_1$ represents the state without generator, and the active power loss is 20.293; $P_2$ represents the state with optimization, and the active power loss is 16.2163. Generator voltage, on-load tap changer ratio are all p.u.. Reactive power compensator capacity units are MVar and active power losses in units MW.

表 Ⅻ  Statistics of Trial Results for IQDE in 30-Bus System Containing DGS

Table Ⅻ  Statistics of Trial Results for IQDE in 30-Bus System Containing DGS

	U1	U2	U5	U8	U11	U13	T11	T12	T15	T36	Q10	Q24
P1	1.06	1.045	1.01	1.01	1.082	1.071	1.06	1.04	0.96	1.04	10	3
P2	1.1	1.0759	1.0431	1.0462	1.1	1.1	0.9	1	1.02	0.9	20	4

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5.   Conclusion

In this paper, an improved quantum differential evolution algorithm (IQDE) has been proposed, which is applied to solve OPRF problem in power system. IQDE hybrid algorithm has the following advantages:

5.0.1   Less Convergence Time

Compared with DE, IQDE can converge faster and better, because DE needs a large population size to avoid premature convergence and IQDE can overcome it. Quantum encoding increases the individual diversity and has better parallelism.

5.0.2   Better Search Ability

The IQDE hybrid algorithm has better search ability than DE, because the accelerating evolutionary operation of onlooker bees improves the local search ability, and the random search operation of the scout bees improves the global ability. When the population size is equal, IQDE can get better solution than DE: better performance in the average losses, the minimum losses and the maximum losses. When the population size is small enough for DE to get good solution, IQDE performs much better; when the population size is large enough for DE, IQDE performs as well as DE.

5.0.3   Effectiveness in System Containing DGs

The introducing of DGs impacts the power flow and voltage of the power system, which affects the robustness and effectiveness of DE. As Fig. 6 shown, DE converged much slower and worse than IQDE. So IQDE is more suitable for solving ORPF problem of system containing DGs.